Minimal equation
$y^2 + (x^3 + x^2 + x + 1)y = -x^3 - x$
Invariants
magma: Conductor(LSeries(C)); Factorization($1);
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\( N \) | = | \( 576 \) | = | \( 2^{6} \cdot 3^{2} \) | |
magma: Discriminant(C); Factorization(Integers()!$1);
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\( \Delta \) | = | \(-576\) | = | \( -1 \cdot 2^{6} \cdot 3^{2} \) |
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | = | \(-544\) | = | \( -1 \cdot 2^{5} \cdot 17 \) |
\( I_4 \) | = | \(7936\) | = | \( 2^{8} \cdot 31 \) |
\( I_6 \) | = | \(-1339392\) | = | \( -1 \cdot 2^{12} \cdot 3 \cdot 109 \) |
\( I_{10} \) | = | \(-2359296\) | = | \( -1 \cdot 2^{18} \cdot 3^{2} \) |
\( J_2 \) | = | \(-68\) | = | \( -1 \cdot 2^{2} \cdot 17 \) |
\( J_4 \) | = | \(110\) | = | \( 2 \cdot 5 \cdot 11 \) |
\( J_6 \) | = | \(36\) | = | \( 2^{2} \cdot 3^{2} \) |
\( J_8 \) | = | \(-3637\) | = | \( -1 \cdot 3637 \) |
\( J_{10} \) | = | \(-576\) | = | \( -1 \cdot 2^{6} \cdot 3^{2} \) |
\( g_1 \) | = | \(22717712/9\) | ||
\( g_2 \) | = | \(540430/9\) | ||
\( g_3 \) | = | \(-289\) |
Automorphism group
magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X)\) | \(\simeq\) | \(C_4 \) | (GAP id : [4,1]) | ||
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | \(D_4 \) | (GAP id : [8,3]) |
Rational points
This curve is locally solvable everywhere.
All rational points: (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 0), (1 : 0 : 0)
Number of rational Weierstrass points: \(0\)
Invariants of the Jacobian:
Analytic rank: \(0\)
2-Selmer rank: \(1\)
Order of Ш*: square
Regulator: 1.0
Real period: 22.396252168185806077664977523
Tamagawa numbers: 1 (p = 2), 1 (p = 3)
Torsion: \(\Z/{10}\Z\)
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $E_2$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition
Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
\(x^{2} - 2\)
Decomposes up to isogeny as the square of the elliptic curve:
Elliptic curve 2.2.8.1-9.1-a3
Endomorphisms
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):\(\End (J_{})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\C\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial \(x^{2} - 2\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):\(\End (J_{\overline{\Q}})\) | \(\simeq\) | a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |