Minimal equation
$y^2 + (x^2 + x)y = x^6 - x^5 - 2x^4 + 4x^3 - 3x + 1$
Invariants
magma: Conductor(LSeries(C)); Factorization($1);
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\( N \) | = | \( 20172 \) | = | \( 2^{2} \cdot 3 \cdot 41^{2} \) | |
magma: Discriminant(C); Factorization(Integers()!$1);
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\( \Delta \) | = | \(-968256\) | = | \( -1 \cdot 2^{6} \cdot 3^{2} \cdot 41^{2} \) |
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | = | \(136\) | = | \( 2^{3} \cdot 17 \) |
\( I_4 \) | = | \(774916\) | = | \( 2^{2} \cdot 23 \cdot 8423 \) |
\( I_6 \) | = | \(-219321592\) | = | \( -1 \cdot 2^{3} \cdot 7 \cdot 613 \cdot 6389 \) |
\( I_{10} \) | = | \(-3965976576\) | = | \( -1 \cdot 2^{18} \cdot 3^{2} \cdot 41^{2} \) |
\( J_2 \) | = | \(17\) | = | \( 17 \) |
\( J_4 \) | = | \(-8060\) | = | \( -1 \cdot 2^{2} \cdot 5 \cdot 13 \cdot 31 \) |
\( J_6 \) | = | \(418896\) | = | \( 2^{4} \cdot 3^{2} \cdot 2909 \) |
\( J_8 \) | = | \(-14460592\) | = | \( -1 \cdot 2^{4} \cdot 83 \cdot 10889 \) |
\( J_{10} \) | = | \(-968256\) | = | \( -1 \cdot 2^{6} \cdot 3^{2} \cdot 41^{2} \) |
\( g_1 \) | = | \(-1419857/968256\) | ||
\( g_2 \) | = | \(9899695/242064\) | ||
\( g_3 \) | = | \(-840701/6724\) |
Automorphism group
magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X)\) | \(\simeq\) | \(V_4 \) | (GAP id : [4,2]) | ||
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | \(V_4 \) | (GAP id : [4,2]) |
Rational points
This curve is locally solvable everywhere.
Known rational points: (-3 : -11 : 2), (-3 : 5 : 2), (-1 : 0 : 1), (0 : -1 : 1), (0 : 1 : 1), (1 : -3 : 2), (1 : -2 : 1), (1 : -1 : 0), (1 : 0 : 1), (1 : 1 : 0), (2 : -9 : 1), (2 : 3 : 1), (3 : -68 : 5), (3 : -52 : 5)
Number of rational Weierstrass points: \(2\)
Invariants of the Jacobian:
Analytic rank*: \(2\)
2-Selmer rank: \(3\)
Order of Ш*: square
Regulator: 0.0178319893453
Real period: 16.482903876584195628030077693
Tamagawa numbers: 4 (p = 2), 2 (p = 3), 1 (p = 41)
Torsion: \(\Z/{2}\Z\)
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 246.a1
Elliptic curve 82.a2
Endomorphisms
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 \(\Q\).