Properties

Label 1062.a.6372.1
Conductor $1062$
Discriminant $6372$
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

Learn more

Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(1062\) \(=\) \( 2 \cdot 3^{2} \cdot 59 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(6372\) \(=\) \( 2^{2} \cdot 3^{3} \cdot 59 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(300\) \(=\)  \( 2^{2} \cdot 3 \cdot 5^{2} \)
\( I_4 \)  \(=\) \(2601\) \(=\)  \( 3^{2} \cdot 17^{2} \)
\( I_6 \)  \(=\) \(306603\) \(=\)  \( 3^{2} \cdot 11 \cdot 19 \cdot 163 \)
\( I_{10} \)  \(=\) \(-815616\) \(=\)  \( - 2^{9} \cdot 3^{3} \cdot 59 \)
\( J_2 \)  \(=\) \(75\) \(=\)  \( 3 \cdot 5^{2} \)
\( J_4 \)  \(=\) \(126\) \(=\)  \( 2 \cdot 3^{2} \cdot 7 \)
\( J_6 \)  \(=\) \(-1024\) \(=\)  \( - 2^{10} \)
\( J_8 \)  \(=\) \(-23169\) \(=\)  \( - 3 \cdot 7723 \)
\( J_{10} \)  \(=\) \(-6372\) \(=\)  \( - 2^{2} \cdot 3^{3} \cdot 59 \)
\( g_1 \)  \(=\) \(-87890625/236\)
\( g_2 \)  \(=\) \(-984375/118\)
\( g_3 \)  \(=\) \(160000/177\)

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

All points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((1 : -2 : 1)\) \((1 : -3 : 2)\) \((1 : -6 : 2)\) \((-3 : 70 : 5)\) \((-3 : -168 : 5)\)
All points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((1 : -2 : 1)\) \((1 : -3 : 2)\) \((1 : -6 : 2)\) \((-3 : 70 : 5)\) \((-3 : -168 : 5)\)
All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : -2 : 1)\)
\((1 : 2 : 1)\) \((1 : -3 : 2)\) \((1 : 3 : 2)\) \((-3 : -238 : 5)\) \((-3 : 238 : 5)\)

magma: [C![-3,-168,5],C![-3,70,5],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-6,2],C![1,-3,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1]]; // minimal model
 
magma: [C![-3,-238,5],C![-3,238,5],C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-3,2],C![1,3,2],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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: \(\Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.008698\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.008698\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 2xz^2 - z^3\) \(0.008698\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + z^3\) \(0\) \(2\)

2-torsion field: 6.0.93987.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.008698 \)
Real period: \( 21.57586 \)
Tamagawa product: \( 4 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.187676 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(3\) \(2\) \(3\) \(2\) \(1 + 3 T + 3 T^{2}\)
\(59\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 8 T + 59 T^{2} )\)

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 \(\Q\).