Properties

Label 930.a.930.1
Conductor 930
Discriminant 930
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

Learn more about

Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![15, -45, 37, 0, -7, -1], R![0, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([15, -45, 37, 0, -7, -1]), R([0, 1, 1]))

$y^2 + (x^2 + x)y = -x^5 - 7x^4 + 37x^2 - 45x + 15$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 930 \)  =  \( 2 \cdot 3 \cdot 5 \cdot 31 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(930\)  =  \( 2 \cdot 3 \cdot 5 \cdot 31 \)

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);

\( I_2 \)  =  \(93192\)  =  \( 2^{3} \cdot 3 \cdot 11 \cdot 353 \)
\( I_4 \)  =  \(956292\)  =  \( 2^{2} \cdot 3 \cdot 79691 \)
\( I_6 \)  =  \(29398820232\)  =  \( 2^{3} \cdot 3 \cdot 19 \cdot 751 \cdot 85847 \)
\( I_{10} \)  =  \(3809280\)  =  \( 2^{13} \cdot 3 \cdot 5 \cdot 31 \)
\( J_2 \)  =  \(11649\)  =  \( 3 \cdot 11 \cdot 353 \)
\( J_4 \)  =  \(5644172\)  =  \( 2^{2} \cdot 1411043 \)
\( J_6 \)  =  \(3640360380\)  =  \( 2^{2} \cdot 3 \cdot 5 \cdot 31 \cdot 1249 \cdot 1567 \)
\( J_8 \)  =  \(2637470125259\)  =  \( 64399 \cdot 40955141 \)
\( J_{10} \)  =  \(930\)  =  \( 2 \cdot 3 \cdot 5 \cdot 31 \)
\( g_1 \)  =  \(71502622649365111083/310\)
\( g_2 \)  =  \(1487013548016809538/155\)
\( g_3 \)  =  \(531176338621566\)
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![1,-1,1],C![1,0,0]];

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

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

Number of rational Weierstrass points: \(2\)

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: 1 (p = 2), 1 (p = 3), 1 (p = 5), 1 (p = 31)

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

Torsion: \(\Z/{2}\Z \times \Z/{4}\Z\)

2-torsion field: \(\Q(\sqrt{2}, \sqrt{465})\)

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 15.a6
  Elliptic curve 62.a3

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 \(\Q\).