Properties

Label 968.a.1936.1
Conductor 968
Discriminant -1936
Mordell-Weil group \(\Z \times \Z/{5}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + y = x^6 - x^4$ (homogenize, simplify)
$y^2 + z^3y = x^6 - x^4z^2$ (dehomogenize, simplify)
$y^2 = 4x^6 - 4x^4 + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 0, 0, -1, 0, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 0, 0, -1, 0, 1], R![1]);
 
sage: X = HyperellipticCurve(R([1, 0, 0, 0, -4, 0, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(968\) = \( 2^{3} \cdot 11^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-1936\) = \( - 2^{4} \cdot 11^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-960\) =  \( - 2^{6} \cdot 3 \cdot 5 \)
\( I_4 \)  = \(22848\) =  \( 2^{6} \cdot 3 \cdot 7 \cdot 17 \)
\( I_6 \)  = \(-7647744\) =  \( - 2^{9} \cdot 3 \cdot 13 \cdot 383 \)
\( I_{10} \)  = \(-7929856\) =  \( - 2^{16} \cdot 11^{2} \)
\( J_2 \)  = \(-120\) =  \( - 2^{3} \cdot 3 \cdot 5 \)
\( J_4 \)  = \(362\) =  \( 2 \cdot 181 \)
\( J_6 \)  = \(1344\) =  \( 2^{6} \cdot 3 \cdot 7 \)
\( J_8 \)  = \(-73081\) =  \( - 107 \cdot 683 \)
\( J_{10} \)  = \(-1936\) =  \( - 2^{4} \cdot 11^{2} \)
\( g_1 \)  = \(1555200000/121\)
\( g_2 \)  = \(39096000/121\)
\( g_3 \)  = \(-1209600/121\)

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

Automorphism group

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

Rational points

All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((-1 : -1 : 1)\) \((1 : -1 : 1)\)

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

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable everywhere.

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.080528\) \(\infty\)
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(5\)

2-torsion field: 6.0.30976.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.080528 \)
Real period: \( 26.98675 \)
Tamagawa product: \( 2 \)
Torsion order:\( 5 \)
Leading coefficient: \( 0.173856 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(4\) \(3\) \(2\) \(1 + 2 T + 2 T^{2}\)
\(11\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 88.a1
  Elliptic curve 11.a3

Endomorphisms of the Jacobian

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 \(\Q\).