Properties

Label 5580.a.33480.1
Conductor 5580
Discriminant 33480
Mordell-Weil group \(\Z \times \Z/{3}\Z\)
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

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

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

Invariants

Conductor: \( N \)  \(=\)  \(5580\) \(=\) \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 31 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(33480\) \(=\) \( 2^{3} \cdot 3^{3} \cdot 5 \cdot 31 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(776\) \(=\)  \( 2^{3} \cdot 97 \)
\( I_4 \)  \(=\) \(41668\) \(=\)  \( 2^{2} \cdot 11 \cdot 947 \)
\( I_6 \)  \(=\) \(3370568\) \(=\)  \( 2^{3} \cdot 31 \cdot 13591 \)
\( I_{10} \)  \(=\) \(137134080\) \(=\)  \( 2^{15} \cdot 3^{3} \cdot 5 \cdot 31 \)
\( J_2 \)  \(=\) \(97\) \(=\)  \( 97 \)
\( J_4 \)  \(=\) \(-42\) \(=\)  \( - 2 \cdot 3 \cdot 7 \)
\( J_6 \)  \(=\) \(7956\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 13 \cdot 17 \)
\( J_8 \)  \(=\) \(192492\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 5347 \)
\( J_{10} \)  \(=\) \(33480\) \(=\)  \( 2^{3} \cdot 3^{3} \cdot 5 \cdot 31 \)
\( g_1 \)  \(=\) \(8587340257/33480\)
\( g_2 \)  \(=\) \(-6388711/5580\)
\( g_3 \)  \(=\) \(2079389/930\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : -1 : 1)\) \((1 : -2 : 1)\) \((12 : -531 : 7)\) \((12 : -2128 : 7)\)

magma: [C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![12,-2128,7],C![12,-531,7]];
 

Number of rational Weierstrass points: \(0\)

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: \(\Z \times \Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.027041\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(3\)

2-torsion field: 6.2.2142720.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.027041 \)
Real period: \( 20.90560 \)
Tamagawa product: \( 9 \)
Torsion order:\( 3 \)
Leading coefficient: \( 0.565311 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 \(\Q\).