Properties

Label 294.a.8232.1
Conductor 294
Discriminant 8232
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-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![-14, -9, 4, 0, -2], R![1, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-14, -9, 4, 0, -2]), R([1, 0, 0, 1]))

$y^2 + (x^3 + 1)y = -2x^4 + 4x^2 - 9x - 14$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 294 \)  =  \( 2 \cdot 3 \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(8232\)  =  \( 2^{3} \cdot 3 \cdot 7^{3} \)

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);

\( I_2 \)  =  \(15272\)  =  \( 2^{3} \cdot 23 \cdot 83 \)
\( I_4 \)  =  \(47140\)  =  \( 2^{2} \cdot 5 \cdot 2357 \)
\( I_6 \)  =  \(237965608\)  =  \( 2^{3} \cdot 239 \cdot 124459 \)
\( I_{10} \)  =  \(33718272\)  =  \( 2^{15} \cdot 3 \cdot 7^{3} \)
\( J_2 \)  =  \(1909\)  =  \( 23 \cdot 83 \)
\( J_4 \)  =  \(151354\)  =  \( 2 \cdot 7 \cdot 19 \cdot 569 \)
\( J_6 \)  =  \(15951264\)  =  \( 2^{5} \cdot 3 \cdot 7^{2} \cdot 3391 \)
\( J_8 \)  =  \(1885732415\)  =  \( 5 \cdot 7^{2} \cdot 7696867 \)
\( J_{10} \)  =  \(8232\)  =  \( 2^{3} \cdot 3 \cdot 7^{3} \)
\( g_1 \)  =  \(25353016669288549/8232\)
\( g_2 \)  =  \(75211396489919/588\)
\( g_3 \)  =  \(49431027484/7\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(V_4 \) (GAP id : [4,2])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(V_4 \) (GAP id : [4,2])

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![-2,3,1],C![-2,4,1],C![1,-1,0],C![1,0,0]];

All rational points: (-2 : 3 : 1), (-2 : 4 : 1), (1 : -1 : 0), (1 : 0 : 0)

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(1\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 1 (p = 2), 1 (p = 3), 3 (p = 7)

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

Torsion: \(\Z/{12}\Z\)

2-torsion field: 8.0.12446784.1

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 14.a6
  Elliptic curve 21.a6

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 \(\Q\).