Properties

Label 9188.a.18376.1
Conductor 9188
Discriminant -18376
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

$y^2 + (x^3 + x + 1)y = x^4 - 2x^2 - x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = x^4z^2 - 2x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 6x^4 + 2x^3 - 7x^2 - 2x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -2, 0, 1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -2, 0, 1]), R([1, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, -2, -7, 2, 6, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(9188\) = \( 2^{2} \cdot 2297 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-18376\) = \( - 2^{3} \cdot 2297 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(456\) =  \( 2^{3} \cdot 3 \cdot 19 \)
\( I_4 \)  = \(28932\) =  \( 2^{2} \cdot 3 \cdot 2411 \)
\( I_6 \)  = \(679944\) =  \( 2^{3} \cdot 3 \cdot 41 \cdot 691 \)
\( I_{10} \)  = \(-75268096\) =  \( - 2^{15} \cdot 2297 \)
\( J_2 \)  = \(57\) =  \( 3 \cdot 19 \)
\( J_4 \)  = \(-166\) =  \( - 2 \cdot 83 \)
\( J_6 \)  = \(4020\) =  \( 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
\( J_8 \)  = \(50396\) =  \( 2^{2} \cdot 43 \cdot 293 \)
\( J_{10} \)  = \(-18376\) =  \( - 2^{3} \cdot 2297 \)
\( g_1 \)  = \(-601692057/18376\)
\( g_2 \)  = \(15371019/9188\)
\( g_3 \)  = \(-3265245/4594\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : -1 : 1)\) \((-2 : -1 : 1)\) \((-1 : 1 : 2)\) \((1 : -2 : 1)\) \((-1 : -4 : 2)\) \((-2 : 10 : 1)\)
\((-28 : -1700 : 45)\) \((-28 : -10773 : 45)\)

magma: [C![-28,-10773,45],C![-28,-1700,45],C![-2,-1,1],C![-2,10,1],C![-1,-4,2],C![-1,0,1],C![-1,1,1],C![-1,1,2],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0]];
 

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
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 - 2z^3\) \(0.250346\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.028186\) \(\infty\)

2-torsion field: 6.4.1176064.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.006962 \)
Real period: \( 22.32998 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.466429 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(3\) \(2\) \(3\) \(1 + T + T^{2}\)
\(2297\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 54 T + 2297 T^{2} )\)

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 \(\Q\).