Properties

Label 961.a.961.1
Conductor 961
Discriminant -961
Mordell-Weil group trivial
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = -x^6 - x^5 - 7x^4 + 74x^3 - 145x^2 + 99x - 33$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^6 - x^5z - 7x^4z^2 + 74x^3z^3 - 145x^2z^4 + 99xz^5 - 33z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 - 4x^5 - 26x^4 + 298x^3 - 579x^2 + 398x - 131$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-33, 99, -145, 74, -7, -1, -1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-33, 99, -145, 74, -7, -1, -1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-131, 398, -579, 298, -26, -4, -3]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

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

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

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

Rational points

This curve has no rational points.

magma: [];
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable except over $\R$.

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);
 

Mordell-Weil group of the Jacobian

Group structure: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));
 

2-torsion field: 6.0.61504.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 0.224643 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.449287 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(31\) \(2\) \(2\) \(1\) \(( 1 - T )^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

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

Endomorphisms of the Jacobian

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 \(\Q\).