Properties

Label 10075.c.654875.1
Conductor $10075$
Discriminant $-654875$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + xy = 5x^5 + 41x^4 + 88x^3 + 16x^2 + x$ (homogenize, simplify)
$y^2 + xz^2y = 5x^5z + 41x^4z^2 + 88x^3z^3 + 16x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 20x^5 + 164x^4 + 352x^3 + 65x^2 + 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 16, 88, 41, 5]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 16, 88, 41, 5], R![0, 1]);
 
sage: X = HyperellipticCurve(R([0, 4, 65, 352, 164, 20]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10075\) \(=\) \( 5^{2} \cdot 13 \cdot 31 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-654875\) \(=\) \( - 5^{3} \cdot 13^{2} \cdot 31 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(144016\) \(=\)  \( 2^{4} \cdot 9001 \)
\( I_4 \)  \(=\) \(182644\) \(=\)  \( 2^{2} \cdot 7 \cdot 11 \cdot 593 \)
\( I_6 \)  \(=\) \(6685568599\) \(=\)  \( 103 \cdot 463 \cdot 140191 \)
\( I_{10} \)  \(=\) \(-2619500\) \(=\)  \( - 2^{2} \cdot 5^{3} \cdot 13^{2} \cdot 31 \)
\( J_2 \)  \(=\) \(72008\) \(=\)  \( 2^{3} \cdot 9001 \)
\( J_4 \)  \(=\) \(216017562\) \(=\)  \( 2 \cdot 3 \cdot 191 \cdot 233 \cdot 809 \)
\( J_6 \)  \(=\) \(864154072025\) \(=\)  \( 5^{2} \cdot 23 \cdot 14851 \cdot 101197 \)
\( J_8 \)  \(=\) \(3890604831488089\) \(=\)  \( 11 \cdot 13 \cdot 30643 \cdot 887870861 \)
\( J_{10} \)  \(=\) \(-654875\) \(=\)  \( - 5^{3} \cdot 13^{2} \cdot 31 \)
\( g_1 \)  \(=\) \(-1935992825145263554592768/654875\)
\( g_2 \)  \(=\) \(-80655002008707170079744/654875\)
\( g_3 \)  \(=\) \(-179230810806977336384/26195\)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 2 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 2 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-4 : 0 : 1)\)

magma: [C![-4,2,1],C![0,0,1],C![1,0,0]]; // minimal model
 
magma: [C![-4,0,1],C![0,0,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(3\)

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: \(\Z/{2}\Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-4 : 2 : 1) - (1 : 0 : 0)\) \(x + 4z\) \(=\) \(0,\) \(y\) \(=\) \(2z^3\) \(0\) \(2\)
\((-4 : 2 : 1) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x + 4z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\((-4 : 2 : 1) - (1 : 0 : 0)\) \(x + 4z\) \(=\) \(0,\) \(y\) \(=\) \(2z^3\) \(0\) \(2\)
\((-4 : 2 : 1) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x + 4z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\((-4 : 0 : 1) - (1 : 0 : 0)\) \(x + 4z\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + 4z^3\) \(0\) \(2\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x (x + 4z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)

2-torsion field: 3.1.620.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(4\)
Regulator: \( 1 \)
Real period: \( 1.535741 \)
Tamagawa product: \( 4 \)
Torsion order:\( 4 \)
Leading coefficient: \( 1.535741 \)
Analytic order of Ш: \( 4 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(5\) \(2\) \(3\) \(2\) \(( 1 - T )^{2}\)
\(13\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 2 T + 13 T^{2} )\)
\(31\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 31 T^{2} )\)

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 \(\Q\).