Properties

Label 1300.a.130000.1
Conductor 1300
Discriminant -130000
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![13, 0, 9, 0, 2], R![0, 1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([13, 0, 9, 0, 2]), R([0, 1, 0, 1]))

$y^2 + (x^3 + x)y = 2x^4 + 9x^2 + 13$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 1300 \)  =  \( 2^{2} \cdot 5^{2} \cdot 13 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-130000\)  =  \( -1 \cdot 2^{4} \cdot 5^{4} \cdot 13 \)

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);

\( I_2 \)  =  \(-18400\)  =  \( -1 \cdot 2^{5} \cdot 5^{2} \cdot 23 \)
\( I_4 \)  =  \(158464\)  =  \( 2^{8} \cdot 619 \)
\( I_6 \)  =  \(-968970496\)  =  \( -1 \cdot 2^{8} \cdot 13 \cdot 23 \cdot 12659 \)
\( I_{10} \)  =  \(-532480000\)  =  \( -1 \cdot 2^{16} \cdot 5^{4} \cdot 13 \)
\( J_2 \)  =  \(-2300\)  =  \( -1 \cdot 2^{2} \cdot 5^{2} \cdot 23 \)
\( J_4 \)  =  \(218766\)  =  \( 2 \cdot 3 \cdot 19^{2} \cdot 101 \)
\( J_6 \)  =  \(-27536704\)  =  \( -1 \cdot 2^{6} \cdot 13 \cdot 23 \cdot 1439 \)
\( J_8 \)  =  \(3868964111\)  =  \( 67 \cdot 1033 \cdot 55901 \)
\( J_{10} \)  =  \(-130000\)  =  \( -1 \cdot 2^{4} \cdot 5^{4} \cdot 13 \)
\( g_1 \)  =  \(6436343000000/13\)
\( g_2 \)  =  \(266172592200/13\)
\( g_3 \)  =  \(1120532032\)
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,-4,1],C![-1,6,1],C![1,-6,1],C![1,-1,0],C![1,0,0],C![1,4,1]];

All rational points: (-1 : -4 : 1), (-1 : 6 : 1), (1 : -6 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 4 : 1)

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: \(2\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 3 (p = 2), 4 (p = 5), 1 (p = 13)

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

Torsion: \(\Z/{6}\Z\)

2-torsion field: 4.0.832.1

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 20.a4
  Elliptic curve 65.a2

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 \(\Q\).