Properties

Label 5641.a.5641.1
Conductor 5641
Discriminant 5641
Mordell-Weil group \(\Z \times \Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

Learn more about

Show commands for: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + y = x^6 - x^5 - x^4 + 2x^3 - x$ (homogenize, simplify)
$y^2 + z^3y = x^6 - x^5z - x^4z^2 + 2x^3z^3 - xz^5$ (dehomogenize, simplify)
$y^2 = 4x^6 - 4x^5 - 4x^4 + 8x^3 - 4x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 2, -1, -1, 1], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 2, -1, -1, 1]), R([1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, -4, 0, 8, -4, -4, 4]))
 

Invariants

Conductor: \( N \)  =  \(5641\) = \( 5641 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(5641\) = \( 5641 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(64\) =  \( 2^{6} \)
\( I_4 \)  = \(-6080\) =  \( - 2^{6} \cdot 5 \cdot 19 \)
\( I_6 \)  = \(-3739136\) =  \( - 2^{9} \cdot 67 \cdot 109 \)
\( I_{10} \)  = \(23105536\) =  \( 2^{12} \cdot 5641 \)
\( J_2 \)  = \(8\) =  \( 2^{3} \)
\( J_4 \)  = \(66\) =  \( 2 \cdot 3 \cdot 11 \)
\( J_6 \)  = \(6352\) =  \( 2^{4} \cdot 397 \)
\( J_8 \)  = \(11615\) =  \( 5 \cdot 23 \cdot 101 \)
\( J_{10} \)  = \(5641\) =  \( 5641 \)
\( g_1 \)  = \(32768/5641\)
\( g_2 \)  = \(33792/5641\)
\( g_3 \)  = \(406528/5641\)

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

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

Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((-1 : -1 : 1)\) \((1 : -1 : 1)\) \((2 : 5 : 1)\) \((2 : -6 : 1)\) \((2 : -153 : 7)\) \((2 : -190 : 7)\)

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],C![2,-190,7],C![2,-153,7],C![2,-6,1],C![2,5,1]];
 

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.203184\) \(\infty\)
\((-1 : 0 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.068593\) \(\infty\)

2-torsion field: 6.2.361024.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.013760 \)
Real period: \( 22.17834 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.305181 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(5641\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 70 T + 5641 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

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

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

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