Minimal equation
Minimal equation
Simplified equation
$y^2 + x^3y = -x^4 - 3x^2 + 12$ | (homogenize, simplify) |
$y^2 + x^3y = -x^4z^2 - 3x^2z^4 + 12z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 4x^4 - 12x^2 + 48$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(3456\) | \(=\) | \( 2^{7} \cdot 3^{3} \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(3456,2),R![1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-442368\) | \(=\) | \( - 2^{14} \cdot 3^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(384\) | \(=\) | \( 2^{7} \cdot 3 \) |
\( I_4 \) | \(=\) | \(2295\) | \(=\) | \( 3^{3} \cdot 5 \cdot 17 \) |
\( I_6 \) | \(=\) | \(331704\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 17 \cdot 271 \) |
\( I_{10} \) | \(=\) | \(54\) | \(=\) | \( 2 \cdot 3^{3} \) |
\( J_2 \) | \(=\) | \(1536\) | \(=\) | \( 2^{9} \cdot 3 \) |
\( J_4 \) | \(=\) | \(73824\) | \(=\) | \( 2^{5} \cdot 3 \cdot 769 \) |
\( J_6 \) | \(=\) | \(-36864\) | \(=\) | \( - 2^{12} \cdot 3^{2} \) |
\( J_8 \) | \(=\) | \(-1376651520\) | \(=\) | \( - 2^{8} \cdot 3^{2} \cdot 5 \cdot 73 \cdot 1637 \) |
\( J_{10} \) | \(=\) | \(442368\) | \(=\) | \( 2^{14} \cdot 3^{3} \) |
\( g_1 \) | \(=\) | \(19327352832\) | ||
\( g_2 \) | \(=\) | \(604766208\) | ||
\( g_3 \) | \(=\) | \(-196608\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
Number of rational Weierstrass points: \(2\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{12}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((2 : -4 : 1) - (1 : 0 : 0)\) | \(z (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + 4z^3\) | \(0\) | \(12\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((2 : -4 : 1) - (1 : 0 : 0)\) | \(z (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + 4z^3\) | \(0\) | \(12\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((2 : 0 : 1) - (1 : 1 : 0)\) | \(z (x - 2z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-x^3 + 8z^3\) | \(0\) | \(12\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 16.84431 \) |
Tamagawa product: | \( 6 \) |
Torsion order: | \( 12 \) |
Leading coefficient: | \( 0.701846 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(7\) | \(14\) | \(6\) | \(1\) | |
\(3\) | \(3\) | \(3\) | \(1\) | \(1 + T\) |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.90.3 | yes |
\(3\) | 3.720.4 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $N(\mathrm{U}(1)\times\mathrm{SU}(2))$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{U}(1)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 96.a
Elliptic curve isogeny class 36.a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(8\) in \(\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q(\sqrt{-3}) \) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \C\) |