Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = x^5 + 52x^4 + 684x^3 + 52x^2 + x$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = x^5z + 52x^4z^2 + 684x^3z^3 + 52x^2z^4 + xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^5 + 209x^4 + 2738x^3 + 209x^2 + 4x$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(460362\) | \(=\) | \( 2 \cdot 3 \cdot 7 \cdot 97 \cdot 113 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(460362\) | \(=\) | \( 2 \cdot 3 \cdot 7 \cdot 97 \cdot 113 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(22140804\) | \(=\) | \( 2^{2} \cdot 3 \cdot 7 \cdot 29 \cdot 61 \cdot 149 \) |
\( I_4 \) | \(=\) | \(117874401\) | \(=\) | \( 3 \cdot 349 \cdot 112583 \) |
\( I_6 \) | \(=\) | \(869518347517617\) | \(=\) | \( 3 \cdot 7 \cdot 1279 \cdot 32373444563 \) |
\( I_{10} \) | \(=\) | \(58926336\) | \(=\) | \( 2^{8} \cdot 3 \cdot 7 \cdot 97 \cdot 113 \) |
\( J_2 \) | \(=\) | \(5535201\) | \(=\) | \( 3 \cdot 7 \cdot 29 \cdot 61 \cdot 149 \) |
\( J_4 \) | \(=\) | \(1276597176500\) | \(=\) | \( 2^{2} \cdot 5^{3} \cdot 41 \cdot 89 \cdot 699697 \) |
\( J_6 \) | \(=\) | \(392564872388814972\) | \(=\) | \( 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 17 \cdot 97 \cdot 113 \cdot 2280028969 \) |
\( J_8 \) | \(=\) | \(135806280790917217394843\) | \(=\) | \( 499 \cdot 386741063 \cdot 703718589439 \) |
\( J_{10} \) | \(=\) | \(460362\) | \(=\) | \( 2 \cdot 3 \cdot 7 \cdot 97 \cdot 113 \) |
\( g_1 \) | \(=\) | \(247427339616283001449764499908381/21922\) | ||
\( g_2 \) | \(=\) | \(5154716410341936035266056323250/10961\) | ||
\( g_3 \) | \(=\) | \(26126351127548847549256806\) |
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: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(9x^2 - 466xz + 9z^2\) | \(=\) | \(0,\) | \(216y\) | \(=\) | \(115405xz^2 - 2277z^3\) | \(11.15945\) | \(\infty\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 26xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(25xz^2 + z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(9x^2 - 466xz + 9z^2\) | \(=\) | \(0,\) | \(216y\) | \(=\) | \(115405xz^2 - 2277z^3\) | \(11.15945\) | \(\infty\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 26xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(25xz^2 + z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(9x^2 - 466xz + 9z^2\) | \(=\) | \(0,\) | \(216y\) | \(=\) | \(x^2z + 230811xz^2 - 4554z^3\) | \(11.15945\) | \(\infty\) |
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + xz^2\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 26xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^2z + 51xz^2 + 2z^3\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\sqrt{42}, \sqrt{10961})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(5\) |
Regulator: | \( 11.15945 \) |
Real period: | \( 1.515596 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 4.228308 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + T + 2 T^{2} )\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 7 T^{2} )\) | |
\(97\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 2 T + 97 T^{2} )\) | |
\(113\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 14 T + 113 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.180.3 | yes |
\(3\) | 3.90.1 | no |
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 791.a
Elliptic curve isogeny class 582.d
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 \(\Q\).