Minimal equation
Minimal equation
Simplified equation
$y^2 = x^6 - 4x^4 + 4x^2 - 1$ | (homogenize, simplify) |
$y^2 = x^6 - 4x^4z^2 + 4x^2z^4 - z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 4x^4 + 4x^2 - 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(1600\) | \(=\) | \( 2^{6} \cdot 5^{2} \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(1600,2),R![1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(409600\) | \(=\) | \( 2^{14} \cdot 5^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(248\) | \(=\) | \( 2^{3} \cdot 31 \) |
\( I_4 \) | \(=\) | \(181\) | \(=\) | \( 181 \) |
\( I_6 \) | \(=\) | \(14873\) | \(=\) | \( 107 \cdot 139 \) |
\( I_{10} \) | \(=\) | \(50\) | \(=\) | \( 2 \cdot 5^{2} \) |
\( J_2 \) | \(=\) | \(992\) | \(=\) | \( 2^{5} \cdot 31 \) |
\( J_4 \) | \(=\) | \(39072\) | \(=\) | \( 2^{5} \cdot 3 \cdot 11 \cdot 37 \) |
\( J_6 \) | \(=\) | \(1945600\) | \(=\) | \( 2^{12} \cdot 5^{2} \cdot 19 \) |
\( J_8 \) | \(=\) | \(100853504\) | \(=\) | \( 2^{8} \cdot 151 \cdot 2609 \) |
\( J_{10} \) | \(=\) | \(409600\) | \(=\) | \( 2^{14} \cdot 5^{2} \) |
\( g_1 \) | \(=\) | \(58632501248/25\) | ||
\( g_2 \) | \(=\) | \(2327987904/25\) | ||
\( g_3 \) | \(=\) | \(4674304\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_4$ | 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: \(\Z/{2}\Z \oplus \Z/{6}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(xz^2\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(xz^2\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1/2 : 0) - (1 : 1/2 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - (1 : -1/2 : 0) - (1 : 1/2 : 0)\) | \(2x^2 - z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(1/2xz^2\) | \(0\) | \(6\) |
2-torsion field: \(\Q(\sqrt{5}) \)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 12.84619 \) |
Tamagawa product: | \( 6 \) |
Torsion order: | \( 12 \) |
Leading coefficient: | \( 0.535257 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(6\) | \(14\) | \(6\) | \(1\) | |
\(5\) | \(2\) | \(2\) | \(1\) | \(( 1 + T )^{2}\) |
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? |
---|---|---|
\(2\) | 2.360.1 | yes |
\(3\) | 3.8640.8 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $J(E_1)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
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 20.a
Elliptic curve isogeny class 80.b
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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |