Properties

Label 100234.a.801872.1
Conductor 100234
Discriminant 801872
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

$y^2 + (x^3 + x)y = x^5 - x^4 - 5x^3 - x^2 + 5x - 3$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = x^5z - x^4z^2 - 5x^3z^3 - x^2z^4 + 5xz^5 - 3z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 2x^4 - 20x^3 - 3x^2 + 20x - 12$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, 5, -1, -5, -1, 1], R![0, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 5, -1, -5, -1, 1]), R([0, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-12, 20, -3, -20, -2, 4, 1]))
 

Invariants

Conductor: \( N \)  =  \(100234\) = \( 2 \cdot 23 \cdot 2179 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(801872\) = \( 2^{4} \cdot 23 \cdot 2179 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(8384\) =  \( 2^{6} \cdot 131 \)
\( I_4 \)  = \(-31616\) =  \( - 2^{7} \cdot 13 \cdot 19 \)
\( I_6 \)  = \(-49935424\) =  \( - 2^{6} \cdot 7 \cdot 11 \cdot 10133 \)
\( I_{10} \)  = \(3284467712\) =  \( 2^{16} \cdot 23 \cdot 2179 \)
\( J_2 \)  = \(1048\) =  \( 2^{3} \cdot 131 \)
\( J_4 \)  = \(46092\) =  \( 2^{2} \cdot 3 \cdot 23 \cdot 167 \)
\( J_6 \)  = \(2655225\) =  \( 3^{2} \cdot 5^{2} \cdot 11801 \)
\( J_8 \)  = \(164550834\) =  \( 2 \cdot 3^{2} \cdot 7 \cdot 1305959 \)
\( J_{10} \)  = \(801872\) =  \( 2^{4} \cdot 23 \cdot 2179 \)
\( g_1 \)  = \(79010794805248/50117\)
\( g_2 \)  = \(144165579648/2179\)
\( g_3 \)  = \(182265264900/50117\)

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (2 : -3 : 1),\, (-2 : 5 : 1),\, (2 : -7 : 1),\, (-3 : 12 : 1),\, (-3 : 18 : 1)\)

magma: [C![-3,12,1],C![-3,18,1],C![-2,5,1],C![1,-1,0],C![1,0,0],C![2,-7,1],C![2,-3,1]];
 

Number of rational Weierstrass points: \(1\)

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
\((2 : -7 : 1) - (1 : 0 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + z^3\) \(1.076362\) \(\infty\)
\((-2 : 5 : 1) - (1 : 0 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 3z^3\) \(0.116714\) \(\infty\)

2-torsion field: 5.1.801872.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.125506 \)
Real period: \( 5.600759 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.405863 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(4\) \(1\) \(2\) \(( 1 + T )( 1 + 2 T^{2} )\)
\(23\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 6 T + 23 T^{2} )\)
\(2179\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 88 T + 2179 T^{2} )\)

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 \(\Q\).