Properties

Label 555.a.8325.1
Conductor $555$
Discriminant $8325$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{10}\Z\)
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

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x + 1)y = 3x^5 - 2x^4 - 4x^3 + x^2 + x$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = 3x^5z - 2x^4z^2 - 4x^3z^3 + x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 12x^5 - 8x^4 - 16x^3 + 5x^2 + 6x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, -4, -2, 3]), R([1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 1, -4, -2, 3], R![1, 1]);
 
sage: X = HyperellipticCurve(R([1, 6, 5, -16, -8, 12]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(555\) \(=\) \( 3 \cdot 5 \cdot 37 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(8325\) \(=\) \( 3^{2} \cdot 5^{2} \cdot 37 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1264\) \(=\)  \( 2^{4} \cdot 79 \)
\( I_4 \)  \(=\) \(18124\) \(=\)  \( 2^{2} \cdot 23 \cdot 197 \)
\( I_6 \)  \(=\) \(6869487\) \(=\)  \( 3 \cdot 2289829 \)
\( I_{10} \)  \(=\) \(33300\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37 \)
\( J_2 \)  \(=\) \(632\) \(=\)  \( 2^{3} \cdot 79 \)
\( J_4 \)  \(=\) \(13622\) \(=\)  \( 2 \cdot 7^{2} \cdot 139 \)
\( J_6 \)  \(=\) \(351361\) \(=\)  \( 351361 \)
\( J_8 \)  \(=\) \(9125317\) \(=\)  \( 9125317 \)
\( J_{10} \)  \(=\) \(8325\) \(=\)  \( 3^{2} \cdot 5^{2} \cdot 37 \)
\( g_1 \)  \(=\) \(100828984082432/8325\)
\( g_2 \)  \(=\) \(3438682756096/8325\)
\( g_3 \)  \(=\) \(140342016064/8325\)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (-1 : -9 : 3)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (-1 : -9 : 3)\)
All points: \((1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 0 : 1),\, (-1 : 0 : 3)\)

magma: [C![-1,-9,3],C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0]]; // minimal model
 
magma: [C![-1,0,3],C![0,-1,1],C![0,1,1],C![1,0,1],C![1,0,0]]; // 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: \(\Z/{2}\Z \times \Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -9 : 3) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (3x + z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(10\)
Generator $D_0$ Height Order
\((-1 : -9 : 3) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (3x + z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(10\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \((x - z) (3x + z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0\) \(10\)

2-torsion field: 3.3.148.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 25.69247 \)
Tamagawa product: \( 4 \)
Torsion order:\( 20 \)
Leading coefficient: \( 0.256924 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(1\) \(2\) \(2\) \(( 1 - T )( 1 + T + 3 T^{2} )\)
\(5\) \(1\) \(2\) \(2\) \(( 1 - T )( 1 + 4 T + 5 T^{2} )\)
\(37\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 37 T^{2} )\)

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 \(\Q\).