Properties

Label 22112.b.353792.1
Conductor 22112
Discriminant 353792
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

$y^2 + y = x^6 - 6x^4 - 9x^3 - 4x^2$ (homogenize, simplify)
$y^2 + z^3y = x^6 - 6x^4z^2 - 9x^3z^3 - 4x^2z^4$ (dehomogenize, simplify)
$y^2 = 4x^6 - 24x^4 - 36x^3 - 16x^2 + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -4, -9, -6, 0, 1], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -4, -9, -6, 0, 1]), R([1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 0, -16, -36, -24, 0, 4]))
 

Invariants

Conductor: \( N \)  =  \(22112\) = \( 2^{5} \cdot 691 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(353792\) = \( 2^{9} \cdot 691 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(672\) =  \( 2^{5} \cdot 3 \cdot 7 \)
\( I_4 \)  = \(71232\) =  \( 2^{6} \cdot 3 \cdot 7 \cdot 53 \)
\( I_6 \)  = \(6339072\) =  \( 2^{9} \cdot 3 \cdot 4127 \)
\( I_{10} \)  = \(1449132032\) =  \( 2^{21} \cdot 691 \)
\( J_2 \)  = \(84\) =  \( 2^{2} \cdot 3 \cdot 7 \)
\( J_4 \)  = \(-448\) =  \( - 2^{6} \cdot 7 \)
\( J_6 \)  = \(7680\) =  \( 2^{9} \cdot 3 \cdot 5 \)
\( J_8 \)  = \(111104\) =  \( 2^{9} \cdot 7 \cdot 31 \)
\( J_{10} \)  = \(353792\) =  \( 2^{9} \cdot 691 \)
\( g_1 \)  = \(8168202/691\)
\( g_2 \)  = \(-518616/691\)
\( g_3 \)  = \(105840/691\)

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
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\)
\((-1 : -3 : 2)\) \((-1 : -5 : 2)\) \((-2 : -8 : 3)\) \((-3 : 9 : 2)\) \((-3 : -17 : 2)\) \((-2 : -19 : 3)\)
\((1 : -51 : 5)\) \((1 : -74 : 5)\)

magma: [C![-3,-17,2],C![-3,9,2],C![-2,-19,3],C![-2,-8,3],C![-1,-5,2],C![-1,-3,2],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-74,5],C![1,-51,5],C![1,-1,0],C![1,1,0]];
 

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
\((-1 : -3 : 2) + (-1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x + z) (2x + z)\) \(=\) \(0,\) \(4y\) \(=\) \(-3xz^2 - 3z^3\) \(0.220586\) \(\infty\)
\((-1 : -1 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0.031601\) \(\infty\)

2-torsion field: 6.2.1415168.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.006891 \)
Real period: \( 15.84805 \)
Tamagawa product: \( 8 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.873731 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(9\) \(5\) \(8\) \(1\)
\(691\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 38 T + 691 T^{2} )\)

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 \(\Q\).