Properties

Label 961.a.961.2
Conductor 961
Discriminant -961
Mordell-Weil group \(\Z/{5}\Z\)
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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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = -x^6 + 2x^5 - 8x^4 + 12x^3 - 18x^2 + 12x - 7$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^6 + 2x^5z - 8x^4z^2 + 12x^3z^3 - 18x^2z^4 + 12xz^5 - 7z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 + 8x^5 - 30x^4 + 50x^3 - 71x^2 + 50x - 27$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-7, 12, -18, 12, -8, 2, -1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-7, 12, -18, 12, -8, 2, -1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-27, 50, -71, 50, -30, 8, -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

\( I_2 \)  = \(-22520\) =  \( - 2^{3} \cdot 5 \cdot 563 \)
\( I_4 \)  = \(2014084\) =  \( 2^{2} \cdot 41 \cdot 12281 \)
\( I_6 \)  = \(-14164636792\) =  \( - 2^{3} \cdot 8627 \cdot 205237 \)
\( I_{10} \)  = \(-3936256\) =  \( - 2^{12} \cdot 31^{2} \)
\( J_2 \)  = \(-2815\) =  \( - 5 \cdot 563 \)
\( J_4 \)  = \(309196\) =  \( 2^{2} \cdot 17 \cdot 4547 \)
\( J_6 \)  = \(-43449708\) =  \( - 2^{2} \cdot 3 \cdot 331 \cdot 10939 \)
\( J_8 \)  = \(6677190401\) =  \( 7 \cdot 953884343 \)
\( J_{10} \)  = \(-961\) =  \( - 31^{2} \)
\( g_1 \)  = \(176763257309509375/961\)
\( g_2 \)  = \(6897140364776500/961\)
\( g_3 \)  = \(344305262376300/961\)

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: \(\Z/{5}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(5\)

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: \( 5.616096 \)
Tamagawa product: \( 1 \)
Torsion order:\( 5 \)
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 \(\Q\).