Properties

Label 10020.a.160320.1
Conductor $10020$
Discriminant $-160320$
Mordell-Weil group \(\Z \times \Z/{3}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = -x^4 - 4x^3 + 7x + 4$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 - 4x^3z^3 + 7xz^5 + 4z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 2x^4 - 14x^3 + x^2 + 30x + 17$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([4, 7, 0, -4, -1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![4, 7, 0, -4, -1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([17, 30, 1, -14, -2, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10020\) \(=\) \( 2^{2} \cdot 3 \cdot 5 \cdot 167 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-160320\) \(=\) \( - 2^{6} \cdot 3 \cdot 5 \cdot 167 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1436\) \(=\)  \( 2^{2} \cdot 359 \)
\( I_4 \)  \(=\) \(278545\) \(=\)  \( 5 \cdot 17 \cdot 29 \cdot 113 \)
\( I_6 \)  \(=\) \(179457607\) \(=\)  \( 7 \cdot 4933 \cdot 5197 \)
\( I_{10} \)  \(=\) \(20520960\) \(=\)  \( 2^{13} \cdot 3 \cdot 5 \cdot 167 \)
\( J_2 \)  \(=\) \(359\) \(=\)  \( 359 \)
\( J_4 \)  \(=\) \(-6236\) \(=\)  \( - 2^{2} \cdot 1559 \)
\( J_6 \)  \(=\) \(-1227984\) \(=\)  \( - 2^{4} \cdot 3 \cdot 25583 \)
\( J_8 \)  \(=\) \(-119933488\) \(=\)  \( - 2^{4} \cdot 53 \cdot 233 \cdot 607 \)
\( J_{10} \)  \(=\) \(160320\) \(=\)  \( 2^{6} \cdot 3 \cdot 5 \cdot 167 \)
\( g_1 \)  \(=\) \(5963102065799/160320\)
\( g_2 \)  \(=\) \(-72132246961/40080\)
\( g_3 \)  \(=\) \(-3297162623/3340\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (-1 : 1 : 1),\, (2 : -5 : 1),\, (2 : -6 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (-1 : 1 : 1),\, (2 : -5 : 1),\, (2 : -6 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (2 : -1 : 1),\, (2 : 1 : 1)\)

magma: [C![-1,0,1],C![-1,1,1],C![1,-1,0],C![1,0,0],C![2,-6,1],C![2,-5,1]]; // minimal model
 
magma: [C![-1,-1,1],C![-1,1,1],C![1,-1,0],C![1,1,0],C![2,-1,1],C![2,1,1]]; // simplified model
 

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
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.113711\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(3\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.113711\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(3\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 + z^3\) \(0.113711\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - xz^2 + z^3\) \(0\) \(3\)

2-torsion field: 6.0.160320.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.113711 \)
Real period: \( 19.20449 \)
Tamagawa product: \( 3 \)
Torsion order:\( 3 \)
Leading coefficient: \( 0.727926 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(6\) \(3\) \(1 + T + T^{2}\)
\(3\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 5 T^{2} )\)
\(167\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 18 T + 167 T^{2} )\)

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 \(\Q\).