Properties

Label 5280.a.633600.1
Conductor 5280
Discriminant -633600
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

$y^2 + (x^2 + 1)y = x^5 + 12x^4 + 5x^3 + 4x^2 + 2x$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = x^5z + 12x^4z^2 + 5x^3z^3 + 4x^2z^4 + 2xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 49x^4 + 20x^3 + 18x^2 + 8x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 4, 5, 12, 1]), R([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 4, 5, 12, 1], R![1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 8, 18, 20, 49, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(5280\) \(=\) \( 2^{5} \cdot 3 \cdot 5 \cdot 11 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-633600\) \(=\) \( - 2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-10432\) \(=\)  \( - 2^{6} \cdot 163 \)
\( I_4 \)  \(=\) \(3710464\) \(=\)  \( 2^{9} \cdot 7247 \)
\( I_6 \)  \(=\) \(-15373442048\) \(=\)  \( - 2^{10} \cdot 15013127 \)
\( I_{10} \)  \(=\) \(-2595225600\) \(=\)  \( - 2^{20} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
\( J_2 \)  \(=\) \(-1304\) \(=\)  \( - 2^{3} \cdot 163 \)
\( J_4 \)  \(=\) \(32200\) \(=\)  \( 2^{3} \cdot 5^{2} \cdot 7 \cdot 23 \)
\( J_6 \)  \(=\) \(7557136\) \(=\)  \( 2^{4} \cdot 19 \cdot 24859 \)
\( J_8 \)  \(=\) \(-2722836336\) \(=\)  \( - 2^{4} \cdot 3 \cdot 11 \cdot 47 \cdot 109721 \)
\( J_{10} \)  \(=\) \(-633600\) \(=\)  \( - 2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
\( g_1 \)  \(=\) \(14728142981504/2475\)
\( g_2 \)  \(=\) \(11156004272/99\)
\( g_3 \)  \(=\) \(-50196386596/2475\)

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
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 2 : 1)\) \((-1 : -4 : 1)\) \((1 : 4 : 1)\)
\((1 : -6 : 1)\) \((-1 : -34 : 4)\) \((-2 : -348 : 9)\) \((-2 : -417 : 9)\)

magma: [C![-2,-417,9],C![-2,-348,9],C![-1,-34,4],C![-1,-4,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-6,1],C![1,0,0],C![1,4,1]];
 

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
\((-1 : 2 : 1) + (0 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 - z^3\) \(0.008231\) \(\infty\)
\((-1 : -34 : 4) - (1 : 0 : 0)\) \(4x + z\) \(=\) \(0,\) \(32y\) \(=\) \(-17z^3\) \(0\) \(2\)

2-torsion field: 4.2.2816.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.008231 \)
Real period: \( 16.18182 \)
Tamagawa product: \( 16 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.532825 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(8\) \(4\) \(1 + T\)
\(3\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + T + 3 T^{2} )\)
\(5\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + T + 5 T^{2} )\)
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 6 T + 11 T^{2} )\)

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 \(\Q\).