Properties

Label 630.a.34020.1
Conductor 630
Discriminant 34020
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![15, -6, -23, 0, 10, 3], R![0, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([15, -6, -23, 0, 10, 3]), R([0, 1, 1]))

$y^2 + (x^2 + x)y = 3x^5 + 10x^4 - 23x^2 - 6x + 15$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 630 \)  =  \( 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(34020\)  =  \( 2^{2} \cdot 3^{5} \cdot 5 \cdot 7 \)

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);

\( I_2 \)  =  \(48200\)  =  \( 2^{3} \cdot 5^{2} \cdot 241 \)
\( I_4 \)  =  \(3879172\)  =  \( 2^{2} \cdot 11 \cdot 131 \cdot 673 \)
\( I_6 \)  =  \(59796026120\)  =  \( 2^{3} \cdot 5 \cdot 1494900653 \)
\( I_{10} \)  =  \(139345920\)  =  \( 2^{14} \cdot 3^{5} \cdot 5 \cdot 7 \)
\( J_2 \)  =  \(6025\)  =  \( 5^{2} \cdot 241 \)
\( J_4 \)  =  \(1472118\)  =  \( 2 \cdot 3 \cdot 73 \cdot 3361 \)
\( J_6 \)  =  \(470090880\)  =  \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 89 \cdot 131 \)
\( J_8 \)  =  \(166291536519\)  =  \( 3^{3} \cdot 331 \cdot 1499 \cdot 12413 \)
\( J_{10} \)  =  \(34020\)  =  \( 2^{2} \cdot 3^{5} \cdot 5 \cdot 7 \)
\( g_1 \)  =  \(1587871127345703125/6804\)
\( g_2 \)  =  \(10732293030978125/1134\)
\( g_3 \)  =  \(13543327580000/27\)
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![-5,-15,3],C![-2,-1,1],C![1,-1,1],C![1,0,0]];

All rational points: (-5 : -15 : 3), (-2 : -1 : 1), (1 : -1 : 1), (1 : 0 : 0)

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

Number of rational Weierstrass points: \(4\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(3\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 2 (p = 2), 2 (p = 3), 1 (p = 5), 1 (p = 7)

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

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

2-torsion field: \(\Q(\sqrt{105}) \)

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 42.a4

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 \(\Q\).