Properties

Label 59107.a.59107.1
Conductor 59107
Discriminant 59107
Mordell-Weil group \(\Z \times \Z \times \Z\)
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + 1)y = 2x^4 - 3x^2 - x$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + z^3)y = 2x^4z^2 - 3x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 9x^4 + 2x^3 - 10x^2 - 4x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -3, 0, 2], R![1, 0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -3, 0, 2]), R([1, 0, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, -4, -10, 2, 9, 2, 1]))
 

Invariants

Conductor: \( N \)  =  \(59107\) = \( 59107 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(59107\) = \( 59107 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(904\) =  \( 2^{3} \cdot 113 \)
\( I_4 \)  = \(81892\) =  \( 2^{2} \cdot 59 \cdot 347 \)
\( I_6 \)  = \(10385896\) =  \( 2^{3} \cdot 149 \cdot 8713 \)
\( I_{10} \)  = \(242102272\) =  \( 2^{12} \cdot 59107 \)
\( J_2 \)  = \(113\) =  \( 113 \)
\( J_4 \)  = \(-321\) =  \( - 3 \cdot 107 \)
\( J_6 \)  = \(12085\) =  \( 5 \cdot 2417 \)
\( J_8 \)  = \(315641\) =  \( 439 \cdot 719 \)
\( J_{10} \)  = \(59107\) =  \( 59107 \)
\( g_1 \)  = \(18424351793/59107\)
\( g_2 \)  = \(-463169937/59107\)
\( g_3 \)  = \(154313365/59107\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\)
\((1 : -1 : 1)\) \((-1 : -1 : 2)\) \((1 : -2 : 1)\) \((-3 : -6 : 1)\) \((-1 : -8 : 2)\) \((-2 : -9 : 3)\)
\((-2 : -22 : 3)\) \((-3 : 23 : 1)\) \((-4 : 36 : 3)\) \((-4 : -47 : 3)\) \((11 : -616 : 10)\) \((11 : -2925 : 10)\)

magma: [C![-4,-47,3],C![-4,36,3],C![-3,-6,1],C![-3,23,1],C![-2,-22,3],C![-2,-9,3],C![-1,-8,2],C![-1,-1,1],C![-1,-1,2],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![11,-2925,10],C![11,-616,10]];
 

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: \(\Z \times \Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.514009\) \(\infty\)
\((-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.232060\) \(\infty\)
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.269174\) \(\infty\)

2-torsion field: 6.2.3782848.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.027565 \)
Real period: \( 20.50443 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.565206 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(59107\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 200 T + 59107 T^{2} )\)

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 \(\Q\).