Properties

Label 762.a.82296.1
Conductor $762$
Discriminant $82296$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{12}\Z\)
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

$y^2 + (x^2 + x)y = x^5 - 8x^4 + 14x^3 + 2x^2 - x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z - 8x^4z^2 + 14x^3z^3 + 2x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 - 31x^4 + 58x^3 + 9x^2 - 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 2, 14, -8, 1]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 2, 14, -8, 1], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([0, -4, 9, 58, -31, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(762\) \(=\) \( 2 \cdot 3 \cdot 127 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(82296\) \(=\) \( 2^{3} \cdot 3^{4} \cdot 127 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(12004\) \(=\)  \( 2^{2} \cdot 3001 \)
\( I_4 \)  \(=\) \(205249\) \(=\)  \( 11 \cdot 47 \cdot 397 \)
\( I_6 \)  \(=\) \(810020577\) \(=\)  \( 3 \cdot 270006859 \)
\( I_{10} \)  \(=\) \(10533888\) \(=\)  \( 2^{10} \cdot 3^{4} \cdot 127 \)
\( J_2 \)  \(=\) \(3001\) \(=\)  \( 3001 \)
\( J_4 \)  \(=\) \(366698\) \(=\)  \( 2 \cdot 183349 \)
\( J_6 \)  \(=\) \(58441312\) \(=\)  \( 2^{5} \cdot 1826291 \)
\( J_8 \)  \(=\) \(10228738527\) \(=\)  \( 3^{2} \cdot 7 \cdot 3089 \cdot 52561 \)
\( J_{10} \)  \(=\) \(82296\) \(=\)  \( 2^{3} \cdot 3^{4} \cdot 127 \)
\( g_1 \)  \(=\) \(243405270090015001/82296\)
\( g_2 \)  \(=\) \(4955375073324349/41148\)
\( g_3 \)  \(=\) \(65790314289164/10287\)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 2 : 1),\, (1 : -4 : 1),\, (4 : -10 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 2 : 1),\, (1 : -4 : 1),\, (4 : -10 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (4 : 0 : 1),\, (1 : -6 : 1),\, (1 : 6 : 1)\)

magma: [C![0,0,1],C![1,-4,1],C![1,0,0],C![1,2,1],C![4,-10,1]]; // minimal model
 
magma: [C![0,0,1],C![1,-6,1],C![1,0,0],C![1,6,1],C![4,0,1]]; // simplified model
 

Number of rational Weierstrass points: \(3\)

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: \(\Z/{2}\Z \times \Z/{12}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) + (4 : -10 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + 4z) x\) \(=\) \(0,\) \(2y\) \(=\) \(-5xz^2\) \(0\) \(2\)
\((0 : 0 : 1) + (1 : 2 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2\) \(0\) \(12\)
Generator $D_0$ Height Order
\((0 : 0 : 1) + (4 : -10 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + 4z) x\) \(=\) \(0,\) \(2y\) \(=\) \(-5xz^2\) \(0\) \(2\)
\((0 : 0 : 1) + (1 : 2 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2\) \(0\) \(12\)
Generator $D_0$ Height Order
\((0 : 0 : 1) - (1 : 0 : 0)\) \((-x + 4z) x\) \(=\) \(0,\) \(2y\) \(=\) \(x^2z - 9xz^2\) \(0\) \(2\)
\((0 : 0 : 1) + (1 : 6 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(x^2z + 5xz^2\) \(0\) \(12\)

2-torsion field: 3.3.1016.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 16.73344 \)
Tamagawa product: \( 12 \)
Torsion order:\( 24 \)
Leading coefficient: \( 0.348613 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(3\) \(3\) \(( 1 - T )( 1 + T + 2 T^{2} )\)
\(3\) \(1\) \(4\) \(4\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(127\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 16 T + 127 T^{2} )\)

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 \(\Q\).