Properties

Label 961.a.961.1
Conductor 961
Discriminant -961
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{RM}\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type yes

Related objects

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-33, 99, -145, 74, -7, -1, -1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-33, 99, -145, 74, -7, -1, -1]), R([1, 1, 0, 1]))
 

$y^2 + (x^3 + x + 1)y = -x^6 - x^5 - 7x^4 + 74x^3 - 145x^2 + 99x - 33$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 961 \)  =  \( 31^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-961\)  =  \( -1 \cdot 31^{2} \)

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);
 

\( I_2 \)  =  \(133960\)  =  \( 2^{3} \cdot 5 \cdot 17 \cdot 197 \)
\( I_4 \)  =  \(4045749124\)  =  \( 2^{2} \cdot 859 \cdot 1177459 \)
\( I_6 \)  =  \(112130831268488\)  =  \( 2^{3} \cdot 199 \cdot 38281 \cdot 1839919 \)
\( I_{10} \)  =  \(-3936256\)  =  \( -1 \cdot 2^{12} \cdot 31^{2} \)
\( J_2 \)  =  \(16745\)  =  \( 5 \cdot 17 \cdot 197 \)
\( J_4 \)  =  \(-30460094\)  =  \( -1 \cdot 2 \cdot 7 \cdot 2175721 \)
\( J_6 \)  =  \(12221475912\)  =  \( 2^{3} \cdot 3^{3} \cdot 101 \cdot 560207 \)
\( J_8 \)  =  \(-180792178085599\)  =  \( -1 \cdot 1447 \cdot 105071 \cdot 1189127 \)
\( J_{10} \)  =  \(-961\)  =  \( -1 \cdot 31^{2} \)
\( g_1 \)  =  \(-1316514841399349215625/961\)
\( g_2 \)  =  \(143016680917998700750/961\)
\( g_3 \)  =  \(-3426841043882137800/961\)
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 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: \(0\)

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

2-Selmer rank: \(1\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: twice a square

Regulator: 1.0

Real period: 0.22464386193802039305666483409

Tamagawa numbers: 1 (p = 31)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.0.61504.1

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition

Simple over \(\overline{\Q}\)

Endomorphisms

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

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)\(\Z [\sqrt{5}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{5}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).