Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = -9x^6 + 16x^5 - 35x^4 + 33x^3 - 35x^2 + 16x - 9$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = -9x^6 + 16x^5z - 35x^4z^2 + 33x^3z^3 - 35x^2z^4 + 16xz^5 - 9z^6$ | (dehomogenize, simplify) |
$y^2 = -35x^6 + 64x^5 - 140x^4 + 134x^3 - 140x^2 + 64x - 35$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(10098\) | \(=\) | \( 2 \cdot 3^{3} \cdot 11 \cdot 17 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-272646\) | \(=\) | \( - 2 \cdot 3^{6} \cdot 11 \cdot 17 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(56004\) | \(=\) | \( 2^{2} \cdot 3 \cdot 13 \cdot 359 \) |
\( I_4 \) | \(=\) | \(288321\) | \(=\) | \( 3 \cdot 11 \cdot 8737 \) |
\( I_6 \) | \(=\) | \(5331417537\) | \(=\) | \( 3 \cdot 19 \cdot 93533641 \) |
\( I_{10} \) | \(=\) | \(143616\) | \(=\) | \( 2^{8} \cdot 3 \cdot 11 \cdot 17 \) |
\( J_2 \) | \(=\) | \(42003\) | \(=\) | \( 3^{2} \cdot 13 \cdot 359 \) |
\( J_4 \) | \(=\) | \(73402380\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 407791 \) |
\( J_6 \) | \(=\) | \(170798965524\) | \(=\) | \( 2^{2} \cdot 3^{4} \cdot 11 \cdot 17 \cdot 2819023 \) |
\( J_8 \) | \(=\) | \(446539889810043\) | \(=\) | \( 3^{4} \cdot 198173 \cdot 27818311 \) |
\( J_{10} \) | \(=\) | \(272646\) | \(=\) | \( 2 \cdot 3^{6} \cdot 11 \cdot 17 \) |
\( g_1 \) | \(=\) | \(179338702480653356667/374\) | ||
\( g_2 \) | \(=\) | \(3730727674118765970/187\) | ||
\( g_3 \) | \(=\) | \(1105214886926046\) |
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);
|
Rational points
Number of rational Weierstrass points: \(0\)
This curve is locally solvable except over $\R$ and $\Q_{2}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(5x^2 - 2xz + 5z^2\) | \(=\) | \(0,\) | \(50y\) | \(=\) | \(21xz^2 - 15z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(5x^2 - 2xz + 5z^2\) | \(=\) | \(0,\) | \(50y\) | \(=\) | \(21xz^2 - 15z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(5x^2 - 2xz + 5z^2\) | \(=\) | \(0,\) | \(50y\) | \(=\) | \(x^3 + 42xz^2 - 29z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + z^3\) | \(0\) | \(2\) |
2-torsion field: 8.0.405705964916736.150
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(3\) |
Regulator: | \( 1 \) |
Real period: | \( 3.494508 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.747254 \) |
Analytic order of Ш: | \( 2 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - T + 2 T^{2} )\) | |
\(3\) | \(3\) | \(6\) | \(4\) | \(1 + T\) | |
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 11 T^{2} )\) | |
\(17\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 17 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.90.6 | 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 153.c
Elliptic curve isogeny class 66.b
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).