Properties

Label 10064.a.10064.1
Conductor $10064$
Discriminant $-10064$
Mordell-Weil group \(\Z\)
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

$y^2 + (x^3 + x)y = -x^4 - 2x^3 + x^2 + 2x + 1$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^4z^2 - 2x^3z^3 + x^2z^4 + 2xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 2x^4 - 8x^3 + 5x^2 + 8x + 4$ (minimize, homogenize)

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

Invariants

Conductor: \( N \)  \(=\)  \(10064\) \(=\) \( 2^{4} \cdot 17 \cdot 37 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-10064\) \(=\) \( - 2^{4} \cdot 17 \cdot 37 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(104\) \(=\)  \( 2^{3} \cdot 13 \)
\( I_4 \)  \(=\) \(9760\) \(=\)  \( 2^{5} \cdot 5 \cdot 61 \)
\( I_6 \)  \(=\) \(284236\) \(=\)  \( 2^{2} \cdot 71059 \)
\( I_{10} \)  \(=\) \(40256\) \(=\)  \( 2^{6} \cdot 17 \cdot 37 \)
\( J_2 \)  \(=\) \(52\) \(=\)  \( 2^{2} \cdot 13 \)
\( J_4 \)  \(=\) \(-1514\) \(=\)  \( - 2 \cdot 757 \)
\( J_6 \)  \(=\) \(-7760\) \(=\)  \( - 2^{4} \cdot 5 \cdot 97 \)
\( J_8 \)  \(=\) \(-673929\) \(=\)  \( - 3^{2} \cdot 103 \cdot 727 \)
\( J_{10} \)  \(=\) \(10064\) \(=\)  \( 2^{4} \cdot 17 \cdot 37 \)
\( g_1 \)  \(=\) \(23762752/629\)
\( g_2 \)  \(=\) \(-13305032/629\)
\( g_3 \)  \(=\) \(-1311440/629\)

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

magma: [C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![0,-2,1],C![0,2,1],C![1,-1,0],C![1,1,0]]; // 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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.034575\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.034575\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - xz^2 - 2z^3\) \(0.034575\) \(\infty\)

2-torsion field: 6.0.161024.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.034575 \)
Real period: \( 22.39443 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.774287 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(4\) \(1\) \(1 + T + 2 T^{2}\)
\(17\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 5 T + 17 T^{2} )\)
\(37\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 2 T + 37 T^{2} )\)

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 \(\Q\).