Properties

Label 3462.a.560844.1
Conductor 3462
Discriminant -560844
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

$y^2 + (x + 1)y = x^5 - 3x^3 + 4x^2 - 3x$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = x^5z - 3x^3z^3 + 4x^2z^4 - 3xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 - 12x^3 + 17x^2 - 10x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 4, -3, 0, 1]), R([1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 4, -3, 0, 1], R![1, 1]);
 
sage: X = HyperellipticCurve(R([1, -10, 17, -12, 0, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(3462\) \(=\) \( 2 \cdot 3 \cdot 577 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-560844\) \(=\) \( - 2^{2} \cdot 3^{5} \cdot 577 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-736\) \(=\)  \( - 2^{5} \cdot 23 \)
\( I_4 \)  \(=\) \(-87296\) \(=\)  \( - 2^{8} \cdot 11 \cdot 31 \)
\( I_6 \)  \(=\) \(9360576\) \(=\)  \( 2^{6} \cdot 3^{3} \cdot 5417 \)
\( I_{10} \)  \(=\) \(-2297217024\) \(=\)  \( - 2^{14} \cdot 3^{5} \cdot 577 \)
\( J_2 \)  \(=\) \(-92\) \(=\)  \( - 2^{2} \cdot 23 \)
\( J_4 \)  \(=\) \(1262\) \(=\)  \( 2 \cdot 631 \)
\( J_6 \)  \(=\) \(5185\) \(=\)  \( 5 \cdot 17 \cdot 61 \)
\( J_8 \)  \(=\) \(-517416\) \(=\)  \( - 2^{3} \cdot 3 \cdot 21559 \)
\( J_{10} \)  \(=\) \(-560844\) \(=\)  \( - 2^{2} \cdot 3^{5} \cdot 577 \)
\( g_1 \)  \(=\) \(1647703808/140211\)
\( g_2 \)  \(=\) \(245676064/140211\)
\( g_3 \)  \(=\) \(-10971460/140211\)

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
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : -1 : 1)\) \((-1 : -3 : 1)\) \((-1 : 3 : 1)\)
\((2 : 3 : 1)\) \((2 : -6 : 1)\) \((17 : -2652 : 16)\) \((17 : -5796 : 16)\)

magma: [C![-1,-3,1],C![-1,3,1],C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0],C![2,-6,1],C![2,3,1],C![17,-5796,16],C![17,-2652,16]];
 

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
\((0 : 0 : 1) + (2 : -6 : 1) - 2 \cdot(1 : 0 : 0)\) \((-x + 2z) x\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2\) \(0.015775\) \(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)

2-torsion field: 4.2.6924.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.015775 \)
Real period: \( 11.41134 \)
Tamagawa product: \( 10 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.450053 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 2 T^{2} )\)
\(3\) \(1\) \(5\) \(5\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(577\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 2 T + 577 T^{2} )\)

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 \(\Q\).