Properties

Label 100315.a.501575.1
Conductor 100315
Discriminant -501575
Mordell-Weil group \(\Z \times \Z\)
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

$y^2 + (x^3 + x + 1)y = 2x^5 + 3x^4 + x^3 + 2x^2 - 5x + 1$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = 2x^5z + 3x^4z^2 + x^3z^3 + 2x^2z^4 - 5xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 8x^5 + 14x^4 + 6x^3 + 9x^2 - 18x + 5$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -5, 2, 1, 3, 2]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -5, 2, 1, 3, 2], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([5, -18, 9, 6, 14, 8, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(100315\) = \( 5 \cdot 20063 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-501575\) = \( - 5^{2} \cdot 20063 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-8760\) =  \( - 2^{3} \cdot 3 \cdot 5 \cdot 73 \)
\( I_4 \)  = \(1011396\) =  \( 2^{2} \cdot 3 \cdot 89 \cdot 947 \)
\( I_6 \)  = \(-3791244216\) =  \( - 2^{3} \cdot 3 \cdot 157968509 \)
\( I_{10} \)  = \(-2054451200\) =  \( - 2^{12} \cdot 5^{2} \cdot 20063 \)
\( J_2 \)  = \(-1095\) =  \( - 3 \cdot 5 \cdot 73 \)
\( J_4 \)  = \(39424\) =  \( 2^{9} \cdot 7 \cdot 11 \)
\( J_6 \)  = \(338316\) =  \( 2^{2} \cdot 3 \cdot 11^{2} \cdot 233 \)
\( J_8 \)  = \(-481176949\) =  \( - 11^{2} \cdot 3976669 \)
\( J_{10} \)  = \(-501575\) =  \( - 5^{2} \cdot 20063 \)
\( g_1 \)  = \(62969549637375/20063\)
\( g_2 \)  = \(2070441838080/20063\)
\( g_3 \)  = \(-16225973676/20063\)

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

Known points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : 1 : 1),\, (1 : -4 : 1),\, (1 : -5 : 2),\, (1 : -8 : 2)\)

magma: [C![1,-8,2],C![1,-5,2],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,1,1]];
 

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(2x^2z\) \(1.038436\) \(\infty\)
\((1 : -4 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 3z^3\) \(0.050193\) \(\infty\)

2-torsion field: 6.4.1284032.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.052092 \)
Real period: \( 14.98942 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.561682 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(5\) \(2\) \(1\) \(2\) \(( 1 - T )( 1 + T + 5 T^{2} )\)
\(20063\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 88 T + 20063 T^{2} )\)

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 \(\Q\).