Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = -18x^6 + 38x^5 - 72x^4 + 73x^3 - 72x^2 + 38x - 18$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = -18x^6 + 38x^5z - 72x^4z^2 + 73x^3z^3 - 72x^2z^4 + 38xz^5 - 18z^6$ | (dehomogenize, simplify) |
$y^2 = -72x^6 + 152x^5 - 287x^4 + 294x^3 - 287x^2 + 152x - 72$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(177660\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 47 \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-355320\) | \(=\) | \( - 2^{3} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 47 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(559644\) | \(=\) | \( 2^{2} \cdot 3 \cdot 149 \cdot 313 \) |
\( I_4 \) | \(=\) | \(461003553\) | \(=\) | \( 3^{2} \cdot 51222617 \) |
\( I_6 \) | \(=\) | \(82699923517551\) | \(=\) | \( 3^{2} \cdot 30161 \cdot 304660999 \) |
\( I_{10} \) | \(=\) | \(45480960\) | \(=\) | \( 2^{10} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 47 \) |
\( J_2 \) | \(=\) | \(139911\) | \(=\) | \( 3 \cdot 149 \cdot 313 \) |
\( J_4 \) | \(=\) | \(796420182\) | \(=\) | \( 2 \cdot 3 \cdot 17 \cdot 137 \cdot 56993 \) |
\( J_6 \) | \(=\) | \(5937657235020\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 47 \cdot 2213 \cdot 45307 \) |
\( J_8 \) | \(=\) | \(49114613777992524\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 647 \cdot 2108647337197 \) |
\( J_{10} \) | \(=\) | \(355320\) | \(=\) | \( 2^{3} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 47 \) |
\( g_1 \) | \(=\) | \(1985617259781851480912613/13160\) | ||
\( g_2 \) | \(=\) | \(40392811153642517443623/6580\) | ||
\( g_3 \) | \(=\) | \(654228089723507037/2\) |
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 $\R$ and $\Q_{2}$.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{6}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(6\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - D_\infty\) | \(x^2 - xz + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - xz^2\) | \(0\) | \(6\) |
2-torsion field: 8.0.6334835525222400.1
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 1 \) |
Real period: | \( 1.506645 \) |
Tamagawa product: | \( 3 \) |
Torsion order: | \( 6 \) |
Leading coefficient: | \( 4.017721 \) |
Analytic order of Ш: | \( 32 \) (rounded) |
Order of Ш: | twice a square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(2\) | \(3\) | \(3\) | \(( 1 - T )^{2}\) | |
\(3\) | \(3\) | \(3\) | \(1\) | \(1 + T\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 + T )( 1 - 2 T + 5 T^{2} )\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 2 T + 7 T^{2} )\) | |
\(47\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 47 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.45.1 | yes |
\(3\) | 3.720.4 | yes |
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 90.b
Elliptic curve isogeny class 1974.g
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 \(\Q\).