Properties

Label 100069.a.100069.1
Conductor 100069
Discriminant 100069
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 1, 0, 2, 1, 1], R![0, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 1, 0, 2, 1, 1]), R([0, 0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-4, 4, 0, 8, 4, 4, 1]))
 

Invariants

Conductor: \( N \)  =  \(100069\) = \( 100069 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(100069\) = \( 100069 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(1984\) =  \( 2^{6} \cdot 31 \)
\( I_4 \)  = \(86848\) =  \( 2^{6} \cdot 23 \cdot 59 \)
\( I_6 \)  = \(60326400\) =  \( 2^{9} \cdot 3 \cdot 5^{2} \cdot 1571 \)
\( I_{10} \)  = \(409882624\) =  \( 2^{12} \cdot 100069 \)
\( J_2 \)  = \(248\) =  \( 2^{3} \cdot 31 \)
\( J_4 \)  = \(1658\) =  \( 2 \cdot 829 \)
\( J_6 \)  = \(-7104\) =  \( - 2^{6} \cdot 3 \cdot 37 \)
\( J_8 \)  = \(-1127689\) =  \( - 563 \cdot 2003 \)
\( J_{10} \)  = \(100069\) =  \( 100069 \)
\( g_1 \)  = \(938120019968/100069\)
\( g_2 \)  = \(25289460736/100069\)
\( g_3 \)  = \(-436924416/100069\)

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (2 : 5 : 1),\, (2 : -13 : 1)\)

magma: [C![1,-1,0],C![1,0,0],C![2,-13,1],C![2,5,1]];
 

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
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.467543\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z\) \(0.492648\) \(\infty\)

2-torsion field: 6.2.6404416.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.210375 \)
Real period: \( 7.311107 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.538079 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(100069\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 6 T + 100069 T^{2} )\)

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 \(\Q\).