Minimal equation
Minimal equation
Simplified equation
$y^2 = x^6 - 5x^4 + 7x^2 - 2$ | (homogenize, simplify) |
$y^2 = x^6 - 5x^4z^2 + 7x^2z^4 - 2z^6$ | (dehomogenize, simplify) |
$y^2 = x^6 - 5x^4 + 7x^2 - 2$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(3200\) | \(=\) | \( 2^{7} \cdot 5^{2} \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(3200,2),R![1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(819200\) | \(=\) | \( 2^{15} \cdot 5^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(520\) | \(=\) | \( 2^{3} \cdot 5 \cdot 13 \) |
\( I_4 \) | \(=\) | \(1141\) | \(=\) | \( 7 \cdot 163 \) |
\( I_6 \) | \(=\) | \(186367\) | \(=\) | \( 227 \cdot 821 \) |
\( I_{10} \) | \(=\) | \(100\) | \(=\) | \( 2^{2} \cdot 5^{2} \) |
\( J_2 \) | \(=\) | \(2080\) | \(=\) | \( 2^{5} \cdot 5 \cdot 13 \) |
\( J_4 \) | \(=\) | \(168096\) | \(=\) | \( 2^{5} \cdot 3 \cdot 17 \cdot 103 \) |
\( J_6 \) | \(=\) | \(17260544\) | \(=\) | \( 2^{13} \cdot 7^{2} \cdot 43 \) |
\( J_8 \) | \(=\) | \(1911416576\) | \(=\) | \( 2^{8} \cdot 7466471 \) |
\( J_{10} \) | \(=\) | \(819200\) | \(=\) | \( 2^{15} \cdot 5^{2} \) |
\( g_1 \) | \(=\) | \(47525504000\) | ||
\( g_2 \) | \(=\) | \(1846534560\) | ||
\( g_3 \) | \(=\) | \(91157248\) |
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: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z/{2}\Z \oplus \Z/{6}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3\) | \(0.717385\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 1 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3\) | \(0.717385\) | \(\infty\) |
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 1 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : -1/2 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(1/2x^3\) | \(0.717385\) | \(\infty\) |
\(D_0 - (1 : -1/2 : 0) - (1 : 1/2 : 0)\) | \(x^2 - xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 1/2 : 1) + (1 : 1/2 : 1) - (1 : -1/2 : 0) - (1 : 1/2 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(1/2z^3\) | \(0\) | \(6\) |
2-torsion field: \(\Q(\sqrt{2}, \sqrt{5})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 0.717385 \) |
Real period: | \( 15.39766 \) |
Tamagawa product: | \( 6 \) |
Torsion order: | \( 12 \) |
Leading coefficient: | \( 0.460252 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(7\) | \(15\) | \(6\) | \(1\) | |
\(5\) | \(2\) | \(2\) | \(1\) | \(( 1 + T )^{2}\) |
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? |
---|---|---|
\(2\) | 2.180.7 | yes |
\(3\) | 3.720.4 | yes |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
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 20.a
Elliptic curve isogeny class 160.a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).