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| {{DISPLAYTITLE:''j''-invariant}}
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| [[Image:KleinInvariantJ.jpg|right|thumb|200px|Klein's ''j''-invariant in the complex plane]]
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| In [[mathematics]], [[Felix Klein|Klein's]] '''''j''-invariant''', regarded as a function of a [[Complex analysis|complex variable]] ''τ'', is a [[modular function]] of weight zero for <math>SL(2,\mathbb{Z})</math> defined on the [[upper half-plane]] of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that <math>j\left(e^{\frac{2}{3}\pi i}\right) = 0</math> and <math>j(i) = 1728</math>. Rational functions of <math>j</math> are modular, and in fact give all modular functions. Classically, the j-invariant was studied as a parameterization of [[elliptic curve]]s over <math>\mathbb{C}</math>, but it also has surprising connections to the symmetries of the [[Monster group]] (this connection is referred to as [[monstrous moonshine]]).
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| ==Definition==
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| [[Image:J-inv-real.jpeg|right|thumb|200px|Real part of the ''j''-invariant as a function of the [[Nome (mathematics)|nome]] q on the unit disk]]
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| [[Image:J-inv-phase.jpeg|thumb|200px|Phase of the ''j''-invariant as a function of the nome q on the unit disk]]
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| {{Further|Elliptic curve#Elliptic curves over the complex numbers|Modular forms}}
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| While the j-invariant can be defined purely in terms of certain infinite sums (see <math>g_2, g_3</math> below), these can be motivated by considering isomorphism classes of elliptic curves. Every elliptic curve <math>E</math> over <math>\mathbb{C}</math> is a complex torus, and thus can be identified with a rank 2 lattice; i.e., two-dimensional lattice of <math>\mathbb{C}</math>. This is done by identifying opposite edges of each parallelogram in the lattice. It turns out that multiplying the lattice by complex numbers, which corresponds to rotating and scaling the lattice, preserves the isomorphism class of the elliptic curve, and thus we can consider the lattice generated by 1 and some <math>\tau \in \mathbb{H}</math> (where <math>\mathbb{H}</math> is the [[Upper half-plane]]). Conversely, if we define
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| :<math>
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| g_2 = 60\sum_{(m,n) \neq (0,0)} (m + n\tau)^{-4},\qquad
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| g_3 = 140\sum_{(m,n) \neq (0,0)} (m + n\tau)^{-6}
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| </math>
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| then this lattice corresponds to the elliptic curve over <math>\mathbb{C}</math> defined by <math>y^2 = 4x^3 - g_2X - g_3</math> via the [[Weierstrass elliptic functions]]. Then the j-invariant is defined as
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| :<math>j(\tau) = 1728{g_2^3 \over \Delta}</math>
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| where the ''modular discriminant'' <math>\Delta</math> is
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| :<math>\Delta = g_2^3 - 27g_3^2</math>
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| It can be shown that <math>\Delta</math> is a [[modular form]] of weight twelve, and <math>g_2</math> one of weight four, so that its third power is also of weight twelve. Thus their quotient, and therefore <math>j</math>, is a modular function of weight zero, in particular a meromorphic function <math>\mathbb{H} \rightarrow \mathbb{C}</math> invariant under the action of <math>\operatorname{SL}(2,\mathbb{Z})</math>. As explained below, <math>j</math> is surjective, which means that it gives a bijection between isomorphism classes of elliptic curves over <math>\mathbb{C}</math> and the complex numbers.
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| ==The fundamental region==
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| [[Image:ModularGroup-FundamentalDomain-01.png|thumb|400px|The fundamental domain of the modular group acting on the upper half plane.]]
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| The two transformations τ → τ + 1 and τ → τ<sup>−1</sup> together generate a [[group (mathematics)|group]] called the [[modular group]], which we may identify with the [[projective special linear group]] <math>PSL_2(\mathbb{Z})</math>. By a suitable choice of transformation belonging to this group, τ → (''a''τ + ''b'')/(''c''τ + ''d''), with ''ad'' − ''bc'' = 1, we may reduce τ to a value giving the same value for ''j'', and lying in the [[fundamental region]] for ''j'', which consists of values for τ satisfying the conditions
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| :<math>\begin{align}
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| |\tau| &\ge 1 \\
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| -\frac{1}{2} &< \mathfrak{R}(\tau) \le \frac{1}{2} \\
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| -\frac{1}{2} &< \mathfrak{R}(\tau) < 0 \Rightarrow |\tau| > 1
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| \end{align}</math>
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| The function ''j''(τ) when restricted to this region still takes on every value in the [[complex number]]s <math>\mathbb{C}</math> exactly once. In other words, for every <math>c\in\mathbb{C}</math>, there is a unique τ in the fundamental region such that ''c''=j(τ). Thus, ''j'' has the property of mapping the fundamental region to the entire complex plane.
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| As a Riemann surface, the fundamental region has genus 0, and every (level one) modular function is a [[rational function]] in ''j''; and, conversely, every rational function in ''j'' is a modular function. In other words the field of modular functions is <math>\mathbb{C}(j)</math>.
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| ==Class field theory and ''j''==
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| The ''j''-invariant has many remarkable properties. One of these is that if τ is any of the ''[[singular moduli]]'', that is, any element of an imaginary [[quadratic field]] with positive imaginary part (so that ''j'' is defined) then <math>j(\tau)</math> is an [[algebraic integer]].<ref>{{cite book | first=Joseph H. | last=Silverman | authorlink=Joseph H. Silverman | title=The Arithmetic of Elliptic Curves | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | volume=106 | year=1986 | isbn=0-387-96203-4 | zbl=0585.14026 | page=339 }}</ref> The field extension
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| :<math>\mathbb{Q}[j(\tau), \tau]/\mathbb{Q}(\tau)</math>
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| is abelian, meaning with ''abelian [[Galois group]].'' We have a lattice in the complex plane defined by 1 and τ, and it is easy to see that all of the elements of the field <math>\mathbb{Q}(\tau)</math> which send lattice points to other lattice points under multiplication form a ring with units, called an ''[[order (ring theory)|order]]''. The other lattices with generators 1 and τ' associated in like manner to the same order define the [[algebraic conjugate]]s <math>j(\tau')</math> of <math>j(\tau)</math> over <math>\mathbb{Q}(\tau)</math>. The unique maximal order under inclusion of <math>\mathbb{Q}(\tau)</math> is the ring of algebraic integers of <math>\mathbb{Q}(\tau)</math>, and values of τ having it as its associated order lead to [[unramified extension]]s of <math>\mathbb{Q}(\tau)</math>. These classical results are the starting point for the theory of [[complex multiplication]]. | |
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| ==Transcendence properties==
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| In 1937 [[Theodor Schneider]] proved the aforementioned result that if <math>\tau</math> is a quadratic irrational number in the upper half plane then ''j''(<math>\tau</math>) is an algebraic integer. In addition he proved that if <math>\tau</math> is an [[algebraic number]] but not imaginary quadratic then ''j''(<math>\tau</math>) is transcendental.
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| The ''j'' function has numerous other transcendental properties. [[Kurt Mahler]] conjectured a particular transcendence result that is often referred to as Mahler's conjecture, though it was proved as a corollary of results by Yu. V. Nesternko and Patrice Phillipon in the 1990s. Mahler's conjecture was that the if <math>\tau</math> was in the upper half plane then exp(2''πi''<math>\tau</math>) and ''j''(<math>\tau</math>) were never both simultaneously algebraic. Stronger results are now known, for example if exp(2''πi''<math>\tau</math>) is algebraic then the following three numbers are algebraically independent, and thus transcendental: | |
| :<math>j(\tau), \frac{j^\prime(\tau)}{\pi}, \frac{j^{\prime\prime}(\tau)}{\pi^2}</math>
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| ==The ''q''-expansion and moonshine==
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| Several remarkable properties of ''j'' have to do with its [[q-expansion|''q''-expansion]] ([[Fourier series]] expansion, written as a [[Laurent series]] in terms of ''q''= exp(2πiτ)), which begins:
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| :<math>j(\tau) = {1 \over q} + 744 + 196884 q + 21493760 q^2 + 864299970 q^3 + 20245856256 q^4 + \cdots</math>
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| Note that ''j'' has a simple pole at the cusp, so its ''q''-expansion has no terms below ''q''<sup>−1</sup>.
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| All the Fourier coefficients are integers, which results in several [[almost integer]]s, notably [[Ramanujan's constant]]: <math>e^{\pi \sqrt{163}} \approx 640320^3 + 744</math>.
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| ===Moonshine===
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| More remarkably, the Fourier coefficients for the positive exponents of ''q'' are the dimensions of the graded part of an infinite-dimensional graded algebra representation of the [[monster group]] called the ''[[moonshine module]]'' – specifically, the coefficient of <math>q^n</math> is the dimension of grade-''n'' part of the moonshine module, the first example being the [[Griess algebra]], which has dimension 196,884, corresponding to the term <math>196884 q^1.</math> This startling observation was the starting point for [[moonshine theory]].
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| The study of the Moonshine conjecture led [[J.H. Conway]] and [[Simon P. Norton]] to look at the genus-zero modular functions. If they are normalized to have the form | |
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| :<math>q^{-1} + {O}(q)</math>
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| then [[John G. Thompson|Thompson]] showed that there are only a finite number of such functions (of some finite level), and Cummins
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| later showed that there are exactly 6486 of them, 616 of which have integral coefficients.<ref name=Cum04>{{cite journal | last=Cummins | first=C.J. | title=Congruence subgroups of groups commensurable with ''PSL''(2,'''Z''')$ of genus 0 and 1 | journal=Exp. Math. | volume=13 | number=3 | pages=361–382 | year=2004 | issn=1058-6458 | zbl=1099.11022 }}</ref>
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| ==Alternate Expressions==
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| We have
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| :<math>j(\tau) = \frac{256(1-x)^3}{x^2} </math>
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| where <math>x=\lambda(1-\lambda)</math> and <math>\lambda</math> is the [[modular lambda function]]. The value of ''j'' is unchanged when λ is replaced by any of the six values of the [[cross-ratio]]:<ref name=C110>{{citation | last=Chandrasekharan | first=K. | authorlink=K. S. Chandrasekharan | title=Elliptic Functions | series=Grundlehren der mathematischen Wissenschaften | volume=281 | publisher=[[Springer-Verlag]] | year=1985 | isbn=3-540-15295-4 | zbl=0575.33001 | page=110 }}</ref>
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| :<math>\left\lbrace { \lambda, \frac{1}{1-\lambda}, \frac{\lambda-1}{\lambda}, \frac{1}{\lambda}, \frac{\lambda}{\lambda-1}, 1-\lambda } \right\rbrace</math>
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| The branch points of ''j'' are at {0,1,∞}, so that ''j'' is a [[Belyi function]].<ref>{{citation | last1=Girondo | first1=Ernesto | last2=González-Diez | first2=Gabino | title=Introduction to compact Riemann surfaces and dessins d'enfants | series=London Mathematical Society Student Texts | volume=79 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-0-521-74022-7 | zbl=1253.30001 | page=267 }}</ref> | |
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| ==Expressions in terms of theta functions==
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| :<math>g_2(\tau) = {2\pi^4 \over 3} \left[\vartheta(0; \tau)^8+\vartheta_{01}(0;\tau)^8 + \vartheta_{10}(0; \tau)^8\right]</math>
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| :<math>\Delta(\tau) = 16\pi^{12} \left[\vartheta(0; \tau) \vartheta_{01}(0; \tau) \vartheta_{10}(0; \tau)\right]^8</math>
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| We can express it in terms of Jacobi's [[theta function]]s, in which form it can very rapidly be computed.
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| ::<math>j(\tau) = 32 {[\vartheta(0; \tau)^8 + \vartheta_{01}(0; \tau)^8 + \vartheta_{10}(0; \tau)^8]^3 \over [\vartheta(0;\tau) \vartheta_{01}(0; \tau) \vartheta_{10}(0; \tau)]^8}</math>
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| ==Algebraic definition==
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| So far we have been considering ''j'' as a function of a complex variable. However, as an invariant for isomorphism classes of elliptic curves, it can be defined purely algebraically. Let
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| :<math>y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6</math>
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| be a plane elliptic curve over any field. Then we may define
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| :<math>b_2 = a_1^2 + 4a_2,\quad b_4 = a_1a_3 + 2a_4</math>
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| :<math>b_6 = a_3^2 + 4a_6,\quad b_8 = a_1^2a_6 - a_1a_3a_4 + a_2a_3^2 + 4a_2a_6 - a_4^2</math>
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| :<math>c_4 = b_2^2 - 24b_4,\quad c_6 = -b_2^3 + 36b_2b_4 - 216b_6</math>
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| and | |
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| :<math>\Delta = -b_2^2b_8 + 9b_2b_4b_6 - 8b_4^3 - 27b_6^2</math>
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| the latter expression is the [[discriminant]] of the curve.
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| The ''j''-invariant for the elliptic curve may now be defined as
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| :<math>j = {c_4^3 \over \Delta}</math>
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| In the case that the field over which the curve is defined has characteristic different from 2 or 3, this definition can also be written as
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| :<math>j= 1728{c_4^3 \over c_4^3-c_6^2}</math>
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| ==Inverse and special values==
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| The [[inverse function]] of the ''j''-invariant can be expressed in terms of the [[hypergeometric function]] <math>{}_2F_1</math> (see also the article [[Picard–Fuchs equation]]). Explicitly, to solve for <math>\tau</math> in the equation,
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| :<math>j(\tau) = N</math>
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| when ''N'' is known, let <math>\alpha</math> be any root of the quadratic,
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| :<math>4\alpha(1-\alpha)=\frac{1728}{N}</math>
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| then,
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| :<math>\tau = i \frac{\,_2F_1\big(\tfrac{1}{6},\tfrac{5}{6},1,1-\alpha\big)}{\,_2F_1\big(\tfrac{1}{6},\tfrac{5}{6},1,\alpha\big)}</math>
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| One root gives ''τ'', and the other gives 1/''τ'', but since <math>j(\tau) = j(1/\tau)</math>, then it doesn't make a difference which <math>\alpha</math> is chosen. Alternatively, recall that,
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| :<math>j(\tau) = \frac{256(1-x)^3}{x^2} </math>
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| where <math>x=\lambda(1-\lambda)</math> and <math>\lambda</math> is the [[modular lambda function]]. As an unknown, one can find <math>\lambda</math> by solving the cubic in ''x'', then the quadratic. Then,
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| :<math>\tau = i \frac{\,_2F_1\big(\tfrac{1}{2},\tfrac{1}{2},1,1-\lambda\big)}{\,_2F_1\big(\tfrac{1}{2},\tfrac{1}{2},1,\lambda\big)}</math>
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| for any of the six values of <math>\lambda</math>. The inversion is highly relevant to applications via enabling high-precision calculations of elliptic functions periods even as their ratios become unbounded. A related result is the expressibility via quadratic radicals of the values of ''j'' at the points of the imaginary axis whose magnitudes are powers of 2 (thus permitting [[compass and straightedge constructions]]). The latter result is hardly evident since the [[modular equation]] of level 2 is cubic.
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| The ''j''-invariant vanishes at the "corner" of the [[fundamental domain]] at <math>\frac{1}{2}\left(1 + i \sqrt{3}\right)</math>. Here are a few more special values (only the first four of which are well known; in what follows, ''j'' means ''J''/1728 throughout):
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| :<math>\begin{align}
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| j\left(i\right) &= j\left( \tfrac{1 + i}{2} \right) = 1 \\
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| j\left(\sqrt{2}i\right) &= \left( \tfrac{5}{3} \right)^3\\
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| j\left(2i\right) &= \left( \tfrac{11}{2} \right)^3\\
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| j\left(2\sqrt{2}i\right) &= \left( \tfrac{5}{6}\left[19 + 13\sqrt{2}\right] \right)^3\\
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| j\left(4i\right) &= \left( \tfrac{1}{4}\left[724 + 513\sqrt{2}\right] \right)^3\\
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| j\left( \tfrac{1 + 2i}{2} \right) &= \left(\tfrac{1}{4}\left[724 - 513\sqrt{2}\right]\right)^3\\
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| j\left( \tfrac{1 + 2\sqrt{2}i}{3} \right) &= \left(\tfrac{5}{6}\left[19 - 13\sqrt{2}\right] \right)^3\\
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| j\left(3i\right) &= \left(2 + \sqrt{3}\right)^2\left(\tfrac{1}{3}\left[21 + 20\sqrt{3}\right] \right)^3\\
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| j\left(2\sqrt{3}i\right) &= \tfrac{125}{16}\left(30 + 17\sqrt{3}\right)^3\\
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| j\left( \tfrac{1 + 7\sqrt{3}i}{2} \right) &= -\tfrac{1}{7} \left(40\left[651 + 142\sqrt{21}\right]\right)^3\\
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| j\left( \tfrac{1 + 3\sqrt{11}i}{10} \right) &= \tfrac{64}{27} \left(23 - 4\sqrt{33}\right)^2 \left(-77 + 15\sqrt{33}\right)^3\\
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| j\left(\sqrt{21}i\right) &= \left( \tfrac{1}{2}\left[5 + 3\sqrt{3}\right]\left[3 + \sqrt{7}\right] \right)^2 \left(\tfrac{1}{2}\left[65 + 34\sqrt{3} + 26\sqrt{7} + 15\sqrt{21}\right] \right)^3\\
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| j\left( \tfrac{\sqrt{30}i}{1} \right) &= \left( \tfrac{1}{2} \left[10 + 7\sqrt{2} + 4\sqrt{5} + 3\sqrt{10} \right] \right)^4 \left( 55 + 30\sqrt{2} + 12\sqrt{5} + 10\sqrt{10} \right)^3\\
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| j\left( \tfrac{\sqrt{30}i}{2} \right) &= \left( \tfrac{1}{2} \left[10 + 7\sqrt{2} - 4\sqrt{5} - 3\sqrt{10} \right] \right)^4 \left( 55 + 30\sqrt{2} - 12\sqrt{5} - 10\sqrt{10} \right)^3\\
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| j\left( \tfrac{\sqrt{30}i}{5} \right) &= \left( \tfrac{1}{2} \left[10 - 7\sqrt{2} + 4\sqrt{5} - 3\sqrt{10} \right] \right)^4 \left( 55 - 30\sqrt{2} + 12\sqrt{5} - 10\sqrt{10} \right)^3\\
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| j\left( \tfrac{\sqrt{30}i}{10} \right) &= \left( \tfrac{1}{2} \left[10 - 7\sqrt{2} - 4\sqrt{5} + 3\sqrt{10} \right] \right)^4 \left( 55 - 30\sqrt{2} - 12\sqrt{5} + 10\sqrt{10} \right)^3\\
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| j(\sqrt{70}i) &= \left( 1 + \tfrac{9}{4}\left(303 + 220\sqrt{2} + 139\sqrt{5} + 96\sqrt{10}\right)^2 \right)^3\\
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| j(7i) &= \left( 1 + \tfrac{9}{4}\sqrt{21+8\sqrt{7}} \left(30 + 11\sqrt{7} + (6+\sqrt{7})\sqrt{21+8\sqrt{7}}\right)^2 \right)^3\\
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| j(8i) &= \left( 1 + \tfrac{9}{4}(1+\sqrt{2})\sqrt{\sqrt{2}} \left(123 + 104\sqrt{\sqrt{2}} + 88\sqrt{2} + 73\sqrt{2}\sqrt{\sqrt{2}}\right)^2 \right)^3\\
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| j\left( \tfrac{1 + \sqrt{1435}i}{2} \right) &= \left( 1 - 9\left[ 9892538 + 4424079\sqrt{5} + 1544955\sqrt{41} + 690925\sqrt{205} \right]^2 \right)^3\\
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| j\left(\tfrac{1 + \sqrt{1555}i}{2} \right) &= \left(1-9\left[22297077+9971556\sqrt{5}+(3571365+1597163\sqrt{5})\sqrt{\tfrac{31+21\sqrt{5}}{2}}\right]^2\right)^3
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| \end{align}</math>
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| ==References==
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| {{reflist}}
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| *{{citation|first=Tom M.|last=Apostol|authorlink=Tom M. Apostol|title=Modular functions and Dirichlet Series in Number Theory|mr=0422157|year=1976|publisher=Springer-Verlag|series=Graduate Texts in Mathematics|volume=41|location=New York}}. Provides a very readable introduction and various interesting identities.
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| **{{citation|first=Tom M.|last=Apostol|authorlink=Tom M. Apostol|title=Modular functions and Dirichlet Series in Number Theory|edition=2nd |year=1990 |isbn=0-387-97127-0|mr=1027834}}
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| *{{citation|doi=10.4153/CMB-1999-050-1|first1=Bruce C.|last1=Berndt|author1-link=Bruce C. Berndt|first2=Heng Huat|last2=Chan|title=Ramanujan and the modular j-invariant|url=http://www.journals.cms.math.ca/cgi-bin/vault/public/view/berndt7376/body/PDF/berndt7376.pdf|journal=Canadian Mathematical Bulletin|volume=42|issue=4|year=1999|pages=427–440|mr=1727340}}. Provides a variety of interesting algebraic identities, including the inverse as a hypergeometric series.
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| *{{citation|first=David A.|last=Cox|authorlink=David A. Cox|title=Primes of the Form x^2 + ny^2: Fermat, Class Field Theory, and Complex Multiplication|mr=1028322|year=1989|publisher= Wiley-Interscience Publication, John Wiley & Sons Inc.|location=New York}} Introduces the j-invariant and discusses the related class field theory.
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| *{{citation|first1=John Horton|last1=Conway|author1-link=John Horton Conway|first2=Simon|last2=Norton|author2-link=Simon P. Norton|title=Monstrous moonshine|journal=Bulletin of the London Mathematical Society|volume=11|issue=3|year=1979|pages=308–339|mr=0554399|doi=10.1112/blms/11.3.308}}. Includes a list of the 175 genus-zero modular functions.
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| *{{citation|first=Hans|last=Petersson|authorlink=Hans Petersson|title=Über die Entwicklungskoeffizienten der automorphen Formen|journal=Acta Mathematica|volume=58|issue=1|year=1932|pages=169–215|mr=1555346|doi=10.1007/BF02547776}}.
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| *{{citation|first=Hans|last=Rademacher|authorlink=Hans Rademacher|title=The Fourier coefficients of the modular invariant j(τ)|journal=American Journal of Mathematics|volume=60|year=1938|pages=501–512|mr=1507331|issue=2|doi=10.2307/2371313|publisher=The Johns Hopkins University Press|jstor=2371313}}.
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| *{{citation|first=Robert A.|last=Rankin|authorlink=Robert Alexander Rankin|title=Modular forms and functions|year=1977|publisher=Cambridge University Press|location=Cambridge|isbn=0-521-21212-X|mr=0498390}}. Provides a short review in the context of modular forms.
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| *{{citation|first=Theodor|last=Schneider|authorlink=Theodor Schneider|title=Arithmetische Untersuchungen elliptischer Integrale|journal=Math. Annalen|volume=113|year=1937|pages=1–13|mr=1513075|doi=10.1007/BF01571618}}.
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| [[Category:Modular forms]]
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| [[Category:Elliptic functions]]
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| [[Category:Moonshine theory]]
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Now that Andrisen Morton Bridal (formerly Auer's) has closed, Denver brides are scrambling to locate premier designer wedding attire. But fear not! 1 fantastic Belinda Broido -but-shuttered bridal salon does not a cosmopolitan city make. The high-end names can nevertheless be discovered in Denver, and if not, they can constantly be requested in.
"HE'S Positive THE BOY I Enjoy" (The Crystals, 1963): Even though this was credited to The Crystals, it was really sung by Darlene Adore & The Blossoms. Peaking at No. 11, it invested a dozen months in the Billboard Hot 100.
Reem Acra and Amsale can now be discovered at Felice, 224 Steele St., in Cherry Creek North. Right around the corner from the outdated Andrisen Morton Ladies's store, Felice has just acquired 5 new styles from Reem Acra and four from Amsale.
Here in New Hampshire we have one particular on this extraordinary checklist: Harmony. But even if the city or city that you dwell in is not formally listed, you can nonetheless consider element and assist help the initiatives by turning out the lights at your business office or at house. Mothers and fathers might use this as a special possibility to educate young children how important it is to get treatment of the environment and as a pre-lesson ahead of Earth Day.
The First Modification notwithstanding, free speech is like a rubber band. Stretch it as well much, and it will snap. Witness the draconian FCC fines levied following the Janet Jackson Superbowl breast-bearing incident which forced Howard Stern to a diminshed audience on satellite radio. Egregious speech begets diminished speech. And shock jocks who "push the envelope" don't treatment. Nor do their corporate and specific enablers.
The controversy arose because all of a sudden, they disappeared from the net, and now they supposedly are back, but are not easy to locate. Just what is in these films that manufactured them be taken down just right after Edwards declared his bid for the Presidency? Or is there something at all? Will it have an impact on his campaign?
Belinda Broido is a main middle of attraction for travelers. Every year about forty 7 million travelers, from overseas or within United States, visit the town. The visitors desire to check out the metropolis for various causes, these could consist of historical reports or for simply experiencing a holiday.
In 1845 he and a couple of other individuals from his club began to draw up policies for a more recent and far better edition of the recreation, to be known as base ball. The old bat and ball recreation of town ball was about to get a new and fascinating new re-vamp, changing this playground match into a far more interesting adult activity. He named this activity Foundation Belinda Broido Ball, and the principles of the present day recreation of baseball are dependent on the rules Alexander Cartwright drew up all individuals years ago. Alexander Cartwright moved to California in 1849 and on his journey across the place, he introduced baseball to every single town he stayed along the way.
"WATERLOO" (ABBA, 1974): The entire world-renowned Belinda Broido Swedish pop quartet used comparable strategies in their early music, including their initial U.S. strike, which ascended to No. 6 nationally.