Minimal equation
$y^2 + x^3y = x^5 - 7x^3 - 16x^2 - 15x - 5$
Invariants
magma: Conductor(LSeries(C)); Factorization($1);
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\( N \) | = | \( 6845 \) | = | \( 5 \cdot 37^{2} \) | |
magma: Discriminant(C); Factorization(Integers()!$1);
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\( \Delta \) | = | \(6845\) | = | \( 5 \cdot 37^{2} \) |
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | = | \(-96\) | = | \( -1 \cdot 2^{5} \cdot 3 \) |
\( I_4 \) | = | \(13632\) | = | \( 2^{6} \cdot 3 \cdot 71 \) |
\( I_6 \) | = | \(900096\) | = | \( 2^{10} \cdot 3 \cdot 293 \) |
\( I_{10} \) | = | \(28037120\) | = | \( 2^{12} \cdot 5 \cdot 37^{2} \) |
\( J_2 \) | = | \(-12\) | = | \( -1 \cdot 2^{2} \cdot 3 \) |
\( J_4 \) | = | \(-136\) | = | \( -1 \cdot 2^{3} \cdot 17 \) |
\( J_6 \) | = | \(-2040\) | = | \( -1 \cdot 2^{3} \cdot 3 \cdot 5 \cdot 17 \) |
\( J_8 \) | = | \(1496\) | = | \( 2^{3} \cdot 11 \cdot 17 \) |
\( J_{10} \) | = | \(6845\) | = | \( 5 \cdot 37^{2} \) |
\( g_1 \) | = | \(-248832/6845\) | ||
\( g_2 \) | = | \(235008/6845\) | ||
\( g_3 \) | = | \(-58752/1369\) |
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 : 7 : 4), (-3 : 20 : 4), (-2 : 3 : 1), (-2 : 5 : 1), (-1 : 0 : 1), (-1 : 1 : 1), (1 : -1 : 0), (1 : 0 : 0), (3 : -20 : 1), (3 : -7 : 1)
Number of rational Weierstrass points: \(0\)
Invariants of the Jacobian:
Analytic rank*: \(2\)
2-Selmer rank: \(2\)
Order of Ш*: square
Regulator: 0.0225460532558
Real period: 14.663072639084885112400475129
Tamagawa numbers: 1 (p = 5), 1 (p = 37)
Torsion: \(\mathrm{trivial}\)
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 37.a1
Elliptic curve 185.b1
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 \(\Q\).