Properties

Label 10025.a.10025.1
Conductor $10025$
Discriminant $10025$
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 for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(10025\) \(=\) \( 5^{2} \cdot 401 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(10025\) \(=\) \( 5^{2} \cdot 401 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(336\) \(=\)  \( 2^{4} \cdot 3 \cdot 7 \)
\( I_4 \)  \(=\) \(1620\) \(=\)  \( 2^{2} \cdot 3^{4} \cdot 5 \)
\( I_6 \)  \(=\) \(219240\) \(=\)  \( 2^{3} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 29 \)
\( I_{10} \)  \(=\) \(-40100\) \(=\)  \( - 2^{2} \cdot 5^{2} \cdot 401 \)
\( J_2 \)  \(=\) \(168\) \(=\)  \( 2^{3} \cdot 3 \cdot 7 \)
\( J_4 \)  \(=\) \(906\) \(=\)  \( 2 \cdot 3 \cdot 151 \)
\( J_6 \)  \(=\) \(-784\) \(=\)  \( - 2^{4} \cdot 7^{2} \)
\( J_8 \)  \(=\) \(-238137\) \(=\)  \( - 3 \cdot 79379 \)
\( J_{10} \)  \(=\) \(-10025\) \(=\)  \( - 5^{2} \cdot 401 \)
\( g_1 \)  \(=\) \(-133827821568/10025\)
\( g_2 \)  \(=\) \(-4295918592/10025\)
\( g_3 \)  \(=\) \(22127616/10025\)

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

magma: [C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,1,0]]; // minimal model
 
magma: [C![0,-1,1],C![0,1,1],C![1,-2,0],C![1,2,0]]; // 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
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0.041216\) \(\infty\)
Generator $D_0$ Height Order
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0.041216\) \(\infty\)
Generator $D_0$ Height Order
\(2 \cdot(0 : 1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2 + z^3\) \(0.041216\) \(\infty\)

2-torsion field: 6.2.641600.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.041216 \)
Real period: \( 22.57242 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.930347 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(5\) \(2\) \(2\) \(1\) \(1 + 4 T + 5 T^{2}\)
\(401\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 30 T + 401 T^{2} )\)

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 \(\Q\).