Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = 5x^6 + 11x^5 + 23x^4 + 23x^3 + 23x^2 + 11x + 5$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = 5x^6 + 11x^5z + 23x^4z^2 + 23x^3z^3 + 23x^2z^4 + 11xz^5 + 5z^6$ | (dehomogenize, simplify) |
$y^2 = 21x^6 + 44x^5 + 92x^4 + 94x^3 + 92x^2 + 44x + 21$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(28050\) | \(=\) | \( 2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-701250\) | \(=\) | \( - 2 \cdot 3 \cdot 5^{4} \cdot 11 \cdot 17 \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(55404\) | \(=\) | \( 2^{2} \cdot 3^{6} \cdot 19 \) |
\( I_4 \) | \(=\) | \(148569\) | \(=\) | \( 3 \cdot 49523 \) |
\( I_6 \) | \(=\) | \(2740841091\) | \(=\) | \( 3^{3} \cdot 101512633 \) |
\( I_{10} \) | \(=\) | \(89760000\) | \(=\) | \( 2^{8} \cdot 3 \cdot 5^{4} \cdot 11 \cdot 17 \) |
\( J_2 \) | \(=\) | \(13851\) | \(=\) | \( 3^{6} \cdot 19 \) |
\( J_4 \) | \(=\) | \(7987568\) | \(=\) | \( 2^{4} \cdot 467 \cdot 1069 \) |
\( J_6 \) | \(=\) | \(6136947300\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17 \cdot 139 \cdot 787 \) |
\( J_8 \) | \(=\) | \(5300403624419\) | \(=\) | \( 67 \cdot 7727 \cdot 10238191 \) |
\( J_{10} \) | \(=\) | \(701250\) | \(=\) | \( 2 \cdot 3 \cdot 5^{4} \cdot 11 \cdot 17 \) |
\( g_1 \) | \(=\) | \(169935608762809431417/233750\) | ||
\( g_2 \) | \(=\) | \(3537583550966246328/116875\) | ||
\( g_3 \) | \(=\) | \(41974138075986/25\) |
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 except over $\Q_{2}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(7x^2 + 3xz + 7z^2\) | \(=\) | \(0,\) | \(49y\) | \(=\) | \(20xz^2 - 35z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(3x^2 + 2xz + 3z^2\) | \(=\) | \(0,\) | \(18y\) | \(=\) | \(5xz^2 - 15z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(7x^2 + 3xz + 7z^2\) | \(=\) | \(0,\) | \(49y\) | \(=\) | \(20xz^2 - 35z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(3x^2 + 2xz + 3z^2\) | \(=\) | \(0,\) | \(18y\) | \(=\) | \(5xz^2 - 15z^3\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(7x^2 + 3xz + 7z^2\) | \(=\) | \(0,\) | \(49y\) | \(=\) | \(x^3 + 40xz^2 - 69z^3\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(3x^2 + 2xz + 3z^2\) | \(=\) | \(0,\) | \(18y\) | \(=\) | \(x^3 + 10xz^2 - 29z^3\) | \(0\) | \(2\) |
2-torsion field: 8.0.405705964916736.151
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1 \) |
Real period: | \( 5.319958 \) |
Tamagawa product: | \( 2 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 2.659979 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - T + 2 T^{2} )\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T^{2} )\) | |
\(5\) | \(2\) | \(4\) | \(2\) | \(( 1 - T )( 1 + T )\) | |
\(11\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 4 T + 11 T^{2} )\) | |
\(17\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 6 T + 17 T^{2} )\) |
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 55.a
Elliptic curve isogeny class 510.f
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 \(\Q\).