Properties

Label 726.a.1452.1
Conductor 726
Discriminant -1452
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![0, -1, -2, 6, 2, 2], R![1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -2, 6, 2, 2]), R([1, 0, 1]))

$y^2 + (x^2 + 1)y = 2x^5 + 2x^4 + 6x^3 - 2x^2 - x$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 726 \)  =  \( 2 \cdot 3 \cdot 11^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-1452\)  =  \( -1 \cdot 2^{2} \cdot 3 \cdot 11^{2} \)

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);

\( I_2 \)  =  \(3040\)  =  \( 2^{5} \cdot 5 \cdot 19 \)
\( I_4 \)  =  \(-1107776\)  =  \( -1 \cdot 2^{6} \cdot 19 \cdot 911 \)
\( I_6 \)  =  \(-1033126976\)  =  \( -1 \cdot 2^{6} \cdot 7^{3} \cdot 19 \cdot 2477 \)
\( I_{10} \)  =  \(-5947392\)  =  \( -1 \cdot 2^{14} \cdot 3 \cdot 11^{2} \)
\( J_2 \)  =  \(380\)  =  \( 2^{2} \cdot 5 \cdot 19 \)
\( J_4 \)  =  \(17556\)  =  \( 2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 19 \)
\( J_6 \)  =  \(702601\)  =  \( 19 \cdot 36979 \)
\( J_8 \)  =  \(-10306189\)  =  \( -1 \cdot 19^{2} \cdot 28549 \)
\( J_{10} \)  =  \(-1452\)  =  \( -1 \cdot 2^{2} \cdot 3 \cdot 11^{2} \)
\( g_1 \)  =  \(-1980879200000/363\)
\( g_2 \)  =  \(-7297976000/11\)
\( g_3 \)  =  \(-25363896100/363\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(C_2 \) (GAP id : [2,1])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(C_2 \) (GAP id : [2,1])

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![0,-1,1],C![0,0,1],C![1,0,0]];

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

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

Number of rational Weierstrass points: \(1\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(1\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: square

Tamagawa numbers: 2 (p = 2), 1 (p = 3), 1 (p = 11)

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

Torsion: \(\Z/{10}\Z\)

2-torsion field: 6.0.52272.1

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 66.c3
  Elliptic curve 11.a3

Endomorphisms

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

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)an order of index \(5\) 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 \(\Q\).