Properties

Label 5904.a.70848.1
Conductor 5904
Discriminant -70848
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(5904\) \(=\) \( 2^{4} \cdot 3^{2} \cdot 41 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-70848\) \(=\) \( - 2^{6} \cdot 3^{3} \cdot 41 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-384\) \(=\)  \( - 2^{7} \cdot 3 \)
\( I_4 \)  \(=\) \(0\) \(=\)  \( 0 \)
\( I_6 \)  \(=\) \(-3502080\) \(=\)  \( - 2^{12} \cdot 3^{2} \cdot 5 \cdot 19 \)
\( I_{10} \)  \(=\) \(-290193408\) \(=\)  \( - 2^{18} \cdot 3^{3} \cdot 41 \)
\( J_2 \)  \(=\) \(-48\) \(=\)  \( - 2^{4} \cdot 3 \)
\( J_4 \)  \(=\) \(96\) \(=\)  \( 2^{5} \cdot 3 \)
\( J_6 \)  \(=\) \(5824\) \(=\)  \( 2^{6} \cdot 7 \cdot 13 \)
\( J_8 \)  \(=\) \(-72192\) \(=\)  \( - 2^{9} \cdot 3 \cdot 47 \)
\( J_{10} \)  \(=\) \(-70848\) \(=\)  \( - 2^{6} \cdot 3^{3} \cdot 41 \)
\( g_1 \)  \(=\) \(147456/41\)
\( g_2 \)  \(=\) \(6144/41\)
\( g_3 \)  \(=\) \(-23296/123\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-2 : 1 : 1)\)
\((1 : -2 : 1)\) \((-2 : 4 : 1)\) \((13 : -2560 : 12)\) \((13 : -5265 : 12)\)

magma: [C![-2,1,1],C![-2,4,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![13,-5265,12],C![13,-2560,12]];
 

Number of rational Weierstrass points: \(2\)

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
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.016344\) \(\infty\)
\((-1 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)

2-torsion field: 4.2.1968.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.016344 \)
Real period: \( 16.82601 \)
Tamagawa product: \( 10 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.687545 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(6\) \(5\) \(1\)
\(3\) \(2\) \(3\) \(2\) \(1 + 2 T + 3 T^{2}\)
\(41\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 41 T^{2} )\)

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 \(\Q\).