The Jacobian of this curve is isogenous to that of the quotient of the modular curve $X_0(33)$ by the Atkin-Lehner involution $w_3$, which has discriminant $3\cdot 11^9$.
Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + 1)y = x^5 + 2x^3 + 4x^2 + 2x$ | (homogenize, simplify) |
$y^2 + (x^2z + z^3)y = x^5z + 2x^3z^3 + 4x^2z^4 + 2xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^5 + x^4 + 8x^3 + 18x^2 + 8x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(363\) | \(=\) | \( 3 \cdot 11^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(-11979\) | \(=\) | \( - 3^{2} \cdot 11^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(344\) | \(=\) | \( 2^{3} \cdot 43 \) |
\( I_4 \) | \(=\) | \(-3068\) | \(=\) | \( - 2^{2} \cdot 13 \cdot 59 \) |
\( I_6 \) | \(=\) | \(-526433\) | \(=\) | \( - 19 \cdot 103 \cdot 269 \) |
\( I_{10} \) | \(=\) | \(-47916\) | \(=\) | \( - 2^{2} \cdot 3^{2} \cdot 11^{3} \) |
\( J_2 \) | \(=\) | \(172\) | \(=\) | \( 2^{2} \cdot 43 \) |
\( J_4 \) | \(=\) | \(1744\) | \(=\) | \( 2^{4} \cdot 109 \) |
\( J_6 \) | \(=\) | \(45841\) | \(=\) | \( 45841 \) |
\( J_8 \) | \(=\) | \(1210779\) | \(=\) | \( 3^{2} \cdot 29 \cdot 4639 \) |
\( J_{10} \) | \(=\) | \(-11979\) | \(=\) | \( - 3^{2} \cdot 11^{3} \) |
\( g_1 \) | \(=\) | \(-150536645632/11979\) | ||
\( g_2 \) | \(=\) | \(-8874253312/11979\) | ||
\( g_3 \) | \(=\) | \(-1356160144/11979\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
Number of rational Weierstrass points: \(3\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{10}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 5xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-15xz^2 - 6z^3\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 5xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-15xz^2 - 6z^3\) | \(0\) | \(10\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : 0 : 1) - (1 : 0 : 0)\) | \(x + z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - z^3\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 - 5xz - 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - 30xz^2 - 11z^3\) | \(0\) | \(10\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 18.97059 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 20 \) |
Leading coefficient: | \( 0.189705 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + T + 3 T^{2} )\) | |
\(11\) | \(2\) | \(3\) | \(2\) | \(( 1 - T )^{2}\) |
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? |
---|---|---|
\(2\) | 2.120.3 | yes |
\(3\) | 3.80.4 | 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 11.a
Elliptic curve isogeny class 33.a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(3\) 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 \(\Q\).