The Jacobian of this curve is isogenous to that of the quotient of the modular curve $X_0(30)$ by the Atkin-Lehner involution $w_2$, which has discriminant $2\cdot 3^9\cdot 5^5$.
Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + 1)y = x^5 + 3x^4 + 3x^3 + 3x^2 + x$ | (homogenize, simplify) |
$y^2 + (x^3 + z^3)y = x^5z + 3x^4z^2 + 3x^3z^3 + 3x^2z^4 + xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 + 12x^4 + 14x^3 + 12x^2 + 4x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(450\) | \(=\) | \( 2 \cdot 3^{2} \cdot 5^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-2700\) | \(=\) | \( - 2^{2} \cdot 3^{3} \cdot 5^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(364\) | \(=\) | \( 2^{2} \cdot 7 \cdot 13 \) |
\( I_4 \) | \(=\) | \(3529\) | \(=\) | \( 3529 \) |
\( I_6 \) | \(=\) | \(393211\) | \(=\) | \( 7 \cdot 13 \cdot 29 \cdot 149 \) |
\( I_{10} \) | \(=\) | \(345600\) | \(=\) | \( 2^{9} \cdot 3^{3} \cdot 5^{2} \) |
\( J_2 \) | \(=\) | \(91\) | \(=\) | \( 7 \cdot 13 \) |
\( J_4 \) | \(=\) | \(198\) | \(=\) | \( 2 \cdot 3^{2} \cdot 11 \) |
\( J_6 \) | \(=\) | \(0\) | \(=\) | \( 0 \) |
\( J_8 \) | \(=\) | \(-9801\) | \(=\) | \( - 3^{4} \cdot 11^{2} \) |
\( J_{10} \) | \(=\) | \(2700\) | \(=\) | \( 2^{2} \cdot 3^{3} \cdot 5^{2} \) |
\( g_1 \) | \(=\) | \(6240321451/2700\) | ||
\( g_2 \) | \(=\) | \(8289281/150\) | ||
\( g_3 \) | \(=\) | \(0\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{24}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(2x^2 + xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(24\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) | \(2x^2 + xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(-xz^2 - z^3\) | \(0\) | \(24\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) | \(2x^2 + xz + z^2\) | \(=\) | \(0,\) | \(2y\) | \(=\) | \(x^3 - 2xz^2 - z^3\) | \(0\) | \(24\) |
2-torsion field: \(\Q(\sqrt{-3}, \sqrt{5})\)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(1\) |
Regulator: | \( 1 \) |
Real period: | \( 18.77899 \) |
Tamagawa product: | \( 6 \) |
Torsion order: | \( 24 \) |
Leading coefficient: | \( 0.195614 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 2 T^{2} )\) | |
\(3\) | \(2\) | \(3\) | \(3\) | \(( 1 - T )( 1 + T )\) | |
\(5\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) |
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? |
---|---|---|
\(2\) | 2.180.4 | 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 15.a
Elliptic curve isogeny class 30.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 \(\Q\).