Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x)y = 26x^6 + 105x^4 + 141x^2 + 63$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2)y = 26x^6 + 105x^4z^2 + 141x^2z^4 + 63z^6$ | (dehomogenize, simplify) |
$y^2 = 105x^6 + 422x^4 + 565x^2 + 252$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(17640\) | \(=\) | \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-105840\) | \(=\) | \( - 2^{4} \cdot 3^{3} \cdot 5 \cdot 7^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(2541320\) | \(=\) | \( 2^{3} \cdot 5 \cdot 63533 \) |
\( I_4 \) | \(=\) | \(84088\) | \(=\) | \( 2^{3} \cdot 23 \cdot 457 \) |
\( I_6 \) | \(=\) | \(71230997940\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 1327 \cdot 894637 \) |
\( I_{10} \) | \(=\) | \(423360\) | \(=\) | \( 2^{6} \cdot 3^{3} \cdot 5 \cdot 7^{2} \) |
\( J_2 \) | \(=\) | \(1270660\) | \(=\) | \( 2^{2} \cdot 5 \cdot 63533 \) |
\( J_4 \) | \(=\) | \(67274020802\) | \(=\) | \( 2 \cdot 163 \cdot 206362027 \) |
\( J_6 \) | \(=\) | \(4749020647640640\) | \(=\) | \( 2^{6} \cdot 3^{3} \cdot 5 \cdot 7^{2} \cdot 89 \cdot 479 \cdot 263129 \) |
\( J_8 \) | \(=\) | \(377149175315781724799\) | \(=\) | \( 107 \cdot 456506213 \cdot 7721162489 \) |
\( J_{10} \) | \(=\) | \(105840\) | \(=\) | \( 2^{4} \cdot 3^{3} \cdot 5 \cdot 7^{2} \) |
\( g_1 \) | \(=\) | \(41405383015594700062115720000/1323\) | ||
\( g_2 \) | \(=\) | \(1725223932914137828957047400/1323\) | ||
\( g_3 \) | \(=\) | \(72445755191484194417600\) |
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: \(\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.890527\) | \(\infty\) |
\(D_0 - D_\infty\) | \(7x^2 + 9z^2\) | \(=\) | \(0,\) | \(7y\) | \(=\) | \(xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(16x^2 + 21z^2\) | \(=\) | \(0,\) | \(16y\) | \(=\) | \(3xz^2\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(z^3\) | \(0.890527\) | \(\infty\) |
\(D_0 - D_\infty\) | \(7x^2 + 9z^2\) | \(=\) | \(0,\) | \(7y\) | \(=\) | \(xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(16x^2 + 21z^2\) | \(=\) | \(0,\) | \(16y\) | \(=\) | \(3xz^2\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + 2z^3\) | \(0.890527\) | \(\infty\) |
\(D_0 - D_\infty\) | \(7x^2 + 9z^2\) | \(=\) | \(0,\) | \(7y\) | \(=\) | \(x^3 + 3xz^2\) | \(0\) | \(2\) |
\(D_0 - D_\infty\) | \(16x^2 + 21z^2\) | \(=\) | \(0,\) | \(16y\) | \(=\) | \(x^3 + 7xz^2\) | \(0\) | \(4\) |
2-torsion field: 8.0.121550625.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(5\) |
Regulator: | \( 0.890527 \) |
Real period: | \( 4.025449 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 0.896193 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(3\) | \(4\) | \(2\) | \(1 + T + 2 T^{2}\) | |
\(3\) | \(2\) | \(3\) | \(2\) | \(( 1 - T )( 1 + T )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 + 2 T + 5 T^{2} )\) | |
\(7\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) |
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 21.a
Elliptic curve isogeny class 840.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 \(\Q\).