Properties

Label 7440.b.89280.1
Conductor 7440
Discriminant -89280
Mordell-Weil group \(\Z \times \Z/{10}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(7440\) \(=\) \( 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-89280\) \(=\) \( - 2^{6} \cdot 3^{2} \cdot 5 \cdot 31 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1664\) \(=\)  \( 2^{7} \cdot 13 \)
\( I_4 \)  \(=\) \(47104\) \(=\)  \( 2^{11} \cdot 23 \)
\( I_6 \)  \(=\) \(20549632\) \(=\)  \( 2^{12} \cdot 29 \cdot 173 \)
\( I_{10} \)  \(=\) \(-365690880\) \(=\)  \( - 2^{18} \cdot 3^{2} \cdot 5 \cdot 31 \)
\( J_2 \)  \(=\) \(208\) \(=\)  \( 2^{4} \cdot 13 \)
\( J_4 \)  \(=\) \(1312\) \(=\)  \( 2^{5} \cdot 41 \)
\( J_6 \)  \(=\) \(13504\) \(=\)  \( 2^{6} \cdot 211 \)
\( J_8 \)  \(=\) \(271872\) \(=\)  \( 2^{9} \cdot 3^{2} \cdot 59 \)
\( J_{10} \)  \(=\) \(-89280\) \(=\)  \( - 2^{6} \cdot 3^{2} \cdot 5 \cdot 31 \)
\( g_1 \)  \(=\) \(-6083264512/1395\)
\( g_2 \)  \(=\) \(-184477696/1395\)
\( g_3 \)  \(=\) \(-9128704/1395\)

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
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((1 : 1 : 1)\)
\((1 : -2 : 1)\) \((-1 : -4 : 2)\) \((20 : 7395 : 9)\) \((20 : -8124 : 9)\)

magma: [C![-1,-4,2],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,0,0],C![1,1,1],C![20,-8124,9],C![20,7395,9]];
 

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: \(\Z \times \Z/{10}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.410806\) \(\infty\)
\((-1 : -4 : 2) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (2x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0\) \(10\)

2-torsion field: 4.2.2480.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.410806 \)
Real period: \( 18.32308 \)
Tamagawa product: \( 10 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.752723 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(6\) \(5\) \(1\)
\(3\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + T + 3 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 5 T^{2} )\)
\(31\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 8 T + 31 T^{2} )\)

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 \(\Q\).