The Jacobian of this curve is isogenous to that of the quotient of the modular curve $X_0(33)$ by the involution $W_3$ (see [MR:1373390]), which has discriminant $3\cdot 11^9$.
Minimal equation
$y^2 + (x^2 + 1)y = x^5 + 2x^3 + 4x^2 + 2x$
Invariants
magma: Conductor(LSeries(C)); Factorization($1);
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\( N \) | = | \( 363 \) | = | \( 3 \cdot 11^{2} \) | |
magma: Discriminant(C); Factorization(Integers()!$1);
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\( \Delta \) | = | \(-11979\) | = | \( -1 \cdot 3^{2} \cdot 11^{3} \) |
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | = | \(1376\) | = | \( 2^{5} \cdot 43 \) |
\( I_4 \) | = | \(-49088\) | = | \( -1 \cdot 2^{6} \cdot 13 \cdot 59 \) |
\( I_6 \) | = | \(-33691712\) | = | \( -1 \cdot 2^{6} \cdot 19 \cdot 103 \cdot 269 \) |
\( I_{10} \) | = | \(-49065984\) | = | \( -1 \cdot 2^{12} \cdot 3^{2} \cdot 11^{3} \) |
\( J_2 \) | = | \(172\) | = | \( 2^{2} \cdot 43 \) |
\( J_4 \) | = | \(1744\) | = | \( 2^{4} \cdot 109 \) |
\( J_6 \) | = | \(45841\) | = | \( 45841 \) |
\( J_8 \) | = | \(1210779\) | = | \( 3^{2} \cdot 29 \cdot 4639 \) |
\( J_{10} \) | = | \(-11979\) | = | \( -1 \cdot 3^{2} \cdot 11^{3} \) |
\( g_1 \) | = | \(-150536645632/11979\) | ||
\( g_2 \) | = | \(-8874253312/11979\) | ||
\( g_3 \) | = | \(-1356160144/11979\) |
Automorphism group
magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X)\) | \(\simeq\) | \(C_2 \) | (GAP id : [2,1]) | ||
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | \(C_2 \) | (GAP id : [2,1]) |
Rational points
This curve is locally solvable everywhere.
All rational points: (-1 : -34 : 4), (-1 : -1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : 0 : 0)
Number of rational Weierstrass points: \(3\)
Invariants of the Jacobian:
Analytic rank: \(0\)
2-Selmer rank: \(2\)
Order of Ш*: square
Regulator: 1.0
Real period: 18.970595995348840324264371503
Tamagawa numbers: 2 (p = 3), 2 (p = 11)
Torsion: \(\Z/{2}\Z \times \Z/{10}\Z\)
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $G_{3,3}$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 11.a3
Elliptic curve 33.a2
Endomorphisms
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 \(\Q\).