Properties

Label 784.c.614656.1
Conductor 784
Discriminant 614656
Sato-Tate group $E_3$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -9, -13, -4, 1], R![]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -9, -13, -4, 1]), R([]))

$y^2 = x^5 - 4x^4 - 13x^3 - 9x^2 - x$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 784 \)  =  \( 2^{4} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(614656\)  =  \( 2^{8} \cdot 7^{4} \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];

G2 invariants

magma: G2Invariants(C);

\( I_2 \)  =  \(6368\)  =  \( 2^{5} \cdot 199 \)
\( I_4 \)  =  \(2308096\)  =  \( 2^{11} \cdot 7^{2} \cdot 23 \)
\( I_6 \)  =  \(3735904256\)  =  \( 2^{13} \cdot 7^{2} \cdot 41 \cdot 227 \)
\( I_{10} \)  =  \(2517630976\)  =  \( 2^{20} \cdot 7^{4} \)
\( J_2 \)  =  \(796\)  =  \( 2^{2} \cdot 199 \)
\( J_4 \)  =  \(2358\)  =  \( 2 \cdot 3^{2} \cdot 131 \)
\( J_6 \)  =  \(-2348\)  =  \( -1 \cdot 2^{2} \cdot 587 \)
\( J_8 \)  =  \(-1857293\)  =  \( -1 \cdot 1857293 \)
\( J_{10} \)  =  \(614656\)  =  \( 2^{8} \cdot 7^{4} \)
\( g_1 \)  =  \(1248318403996/2401\)
\( g_2 \)  =  \(9291226221/4802\)
\( g_3 \)  =  \(-23245787/9604\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(C_6 \) (GAP id : [6,2])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(D_6 \) (GAP id : [12,4])

Rational points

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);

This curve is locally solvable everywhere.

magma: [C![-1,0,1],C![0,0,1],C![1,0,0]];

All rational points: (-1 : 0 : 1), (0 : 0 : 1), (1 : 0 : 0)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

Number of rational Weierstrass points: \(3\)

Invariants of the Jacobian:

Analytic rank: \(0\)

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 3 (p = 2), 3 (p = 7)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: \(\Z/{2}\Z \times \Z/{6}\Z\)

2-torsion field: \(\Q(\zeta_{7})^+\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $E_3$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{7})^+\) with defining polynomial:
  \(x^{3} - x^{2} - 2 x + 1\)

Decomposes up to isogeny as the square of the elliptic curve:
  Elliptic curve 3.3.49.1-64.1-a3

Endomorphisms

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{7})^+\) with defining polynomial \(x^{3} - x^{2} - 2 x + 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\)\(\simeq\)an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)