Minimal equation
$y^2 + x^3y = 3x^4 + 13x^2 + 20$
Invariants
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(640,2),R![1]>*])); Factorization($1);
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\( N \) | = | \( 640 \) | = | \( 2^{7} \cdot 5 \) | |
magma: Discriminant(C); Factorization(Integers()!$1);
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\( \Delta \) | = | \(-81920\) | = | \( -1 \cdot 2^{14} \cdot 5 \) |
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | = | \(-29184\) | = | \( -1 \cdot 2^{9} \cdot 3 \cdot 19 \) |
\( I_4 \) | = | \(150528\) | = | \( 2^{10} \cdot 3 \cdot 7^{2} \) |
\( I_6 \) | = | \(-1460207616\) | = | \( -1 \cdot 2^{16} \cdot 3 \cdot 7 \cdot 1061 \) |
\( I_{10} \) | = | \(-335544320\) | = | \( -1 \cdot 2^{26} \cdot 5 \) |
\( J_2 \) | = | \(-3648\) | = | \( -1 \cdot 2^{6} \cdot 3 \cdot 19 \) |
\( J_4 \) | = | \(552928\) | = | \( 2^{5} \cdot 37 \cdot 467 \) |
\( J_6 \) | = | \(-111431680\) | = | \( -1 \cdot 2^{12} \cdot 5 \cdot 5441 \) |
\( J_8 \) | = | \(25193348864\) | = | \( 2^{8} \cdot 73 \cdot 379 \cdot 3557 \) |
\( J_{10} \) | = | \(-81920\) | = | \( -1 \cdot 2^{14} \cdot 5 \) |
\( g_1 \) | = | \(39432490647552/5\) | ||
\( g_2 \) | = | \(1638374321664/5\) | ||
\( g_3 \) | = | \(18102076416\) |
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.
All rational points: (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: 7.4056741771287726282328124212
Tamagawa numbers: 6 (p = 2), 1 (p = 5)
Torsion: \(\Z/{12}\Z\)
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $N(G_{1,3})$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{U}(1)\times\mathrm{SU}(2)\) |
Decomposition
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 32.a4
Elliptic curve 20.a4
Endomorphisms
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(4\) in \(\Z \times \Z [\sqrt{-1}]\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q(\sqrt{-1}) \) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \C\) |