Properties

Label 1050.a.131250.1
Conductor 1050
Discriminant -131250
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![3, 8, 15, 17, 15, 8, 3], R![0, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 8, 15, 17, 15, 8, 3]), R([0, 1, 1]))

$y^2 + (x^2 + x)y = 3x^6 + 8x^5 + 15x^4 + 17x^3 + 15x^2 + 8x + 3$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 1050 \)  =  \( 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-131250\)  =  \( -1 \cdot 2 \cdot 3 \cdot 5^{5} \cdot 7 \)

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);

\( I_2 \)  =  \(-23736\)  =  \( -1 \cdot 2^{3} \cdot 3 \cdot 23 \cdot 43 \)
\( I_4 \)  =  \(794436\)  =  \( 2^{2} \cdot 3 \cdot 239 \cdot 277 \)
\( I_6 \)  =  \(-6073742904\)  =  \( -1 \cdot 2^{3} \cdot 3 \cdot 12107 \cdot 20903 \)
\( I_{10} \)  =  \(-537600000\)  =  \( -1 \cdot 2^{13} \cdot 3 \cdot 5^{5} \cdot 7 \)
\( J_2 \)  =  \(-2967\)  =  \( -1 \cdot 3 \cdot 23 \cdot 43 \)
\( J_4 \)  =  \(358520\)  =  \( 2^{3} \cdot 5 \cdot 8963 \)
\( J_6 \)  =  \(-56735700\)  =  \( -1 \cdot 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 27017 \)
\( J_8 \)  =  \(9949557875\)  =  \( 5^{3} \cdot 79596463 \)
\( J_{10} \)  =  \(-131250\)  =  \( -1 \cdot 2 \cdot 3 \cdot 5^{5} \cdot 7 \)
\( g_1 \)  =  \(76641937806559869/43750\)
\( g_2 \)  =  \(312136655012892/4375\)
\( g_3 \)  =  \(475666111026/125\)
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 except over $\Q_{5}$.

magma: [];

There are no rational points.

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: \(3\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: twice a square

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

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

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

2-torsion field: 8.0.497871360000.3

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 70.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 \(\Q\).