Properties

Label 15360.f.983040.2
Conductor 15360
Discriminant -983040
Sato-Tate group $N(G_{1,3})$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-30, 0, -37, 0, -15, 0, -2], R![]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-30, 0, -37, 0, -15, 0, -2]), R([]))

$y^2 = -2x^6 - 15x^4 - 37x^2 - 30$

Invariants

magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(15360,2),R![1]>*])); Factorization($1);
\( N \)  =  \( 15360 \)  =  \( 2^{10} \cdot 3 \cdot 5 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-983040\)  =  \( -1 \cdot 2^{16} \cdot 3 \cdot 5 \)

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);

\( I_2 \)  =  \(-372480\)  =  \( -1 \cdot 2^{8} \cdot 3 \cdot 5 \cdot 97 \)
\( I_4 \)  =  \(918528\)  =  \( 2^{10} \cdot 3 \cdot 13 \cdot 23 \)
\( I_6 \)  =  \(-114034114560\)  =  \( -1 \cdot 2^{15} \cdot 3 \cdot 5 \cdot 232003 \)
\( I_{10} \)  =  \(-4026531840\)  =  \( -1 \cdot 2^{28} \cdot 3 \cdot 5 \)
\( J_2 \)  =  \(-46560\)  =  \( -1 \cdot 2^{5} \cdot 3 \cdot 5 \cdot 97 \)
\( J_4 \)  =  \(90316832\)  =  \( 2^{5} \cdot 113 \cdot 24977 \)
\( J_6 \)  =  \(-233570058240\)  =  \( -1 \cdot 2^{14} \cdot 3 \cdot 5 \cdot 19 \cdot 50021 \)
\( J_8 \)  =  \(679472942284544\)  =  \( 2^{8} \cdot 53 \cdot 20921 \cdot 2393723 \)
\( J_{10} \)  =  \(-983040\)  =  \( -1 \cdot 2^{16} \cdot 3 \cdot 5 \)
\( g_1 \)  =  \(222583859461440000\)
\( g_2 \)  =  \(9273345076342800\)
\( g_3 \)  =  \(515076721401600\)
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; HasPointsLocallyEverywhere(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);

This curve is locally solvable except over $\R$.

magma: [];

There are no rational points

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

Number of rational Weierstrass points:

\(0\)

Invariants of the Jacobian:

Analytic rank:

\(1\)

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

2-Selmer rank:

\(4\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*:

twice a square

Tamagawa numbers:

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

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

Torsion:

\(\Z/{2}\Z \times \Z/{2}\Z\)

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 480.b3
  Elliptic curve 32.a3

Endomorphisms

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

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

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 \(\Z \times \Z [\sqrt{-1}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-1}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)