Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = x^5 + 28x^4 + 201x^3 + 28x^2 + x$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = x^5z + 28x^4z^2 + 201x^3z^3 + 28x^2z^4 + xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^5 + 113x^4 + 806x^3 + 113x^2 + 4x$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(2730\) | \(=\) | \( 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(38220\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(1847076\) | \(=\) | \( 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 1999 \) |
\( I_4 \) | \(=\) | \(9790593\) | \(=\) | \( 3 \cdot 971 \cdot 3361 \) |
\( I_6 \) | \(=\) | \(6017800236609\) | \(=\) | \( 3 \cdot 7 \cdot 97 \cdot 7349 \cdot 401993 \) |
\( I_{10} \) | \(=\) | \(4892160\) | \(=\) | \( 2^{9} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \) |
\( J_2 \) | \(=\) | \(461769\) | \(=\) | \( 3 \cdot 7 \cdot 11 \cdot 1999 \) |
\( J_4 \) | \(=\) | \(8884200782\) | \(=\) | \( 2 \cdot 17 \cdot 31 \cdot 311 \cdot 27103 \) |
\( J_6 \) | \(=\) | \(227893017162720\) | \(=\) | \( 2^{5} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \cdot 191 \cdot 193 \cdot 20219 \) |
\( J_8 \) | \(=\) | \(6576226776830660039\) | \(=\) | \( 14341843 \cdot 458534288573 \) |
\( J_{10} \) | \(=\) | \(38220\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \) |
\( g_1 \) | \(=\) | \(142825757240820183004850067/260\) | ||
\( g_2 \) | \(=\) | \(2975399985891326799051477/130\) | ||
\( g_3 \) | \(=\) | \(1271422473017363079336\) |
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: \(\Z/{2}\Z \oplus \Z/{4}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 14xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(13xz^2 + z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 14xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-113xz^2 - 8z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 14xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(13xz^2 + z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 14xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-113xz^2 - 8z^3\) | \(0\) | \(4\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 14xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^2z + 27xz^2 + 2z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 14xz - z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - 225xz^2 - 16z^3\) | \(0\) | \(4\) |
2-torsion field: \(\Q(\sqrt{3}, \sqrt{65})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1 \) |
Real period: | \( 2.823465 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 0.705866 \) |
Analytic order of Ш: | \( 4 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 - T )( 1 + T + 2 T^{2} )\) | |
\(3\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 5 T^{2} )\) | |
\(7\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 7 T^{2} )\) | |
\(13\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 13 T^{2} )\) |
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 130.b
Elliptic curve isogeny class 21.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 \(\Q\).