Properties

Label 1145.a.143125.1
Conductor 1145
Discriminant -143125
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
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

$y^2 + (x^3 + x^2 + x)y = 2x^4 + 4x^3 + 9x^2 + 10x + 9$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2)y = 2x^4z^2 + 4x^3z^3 + 9x^2z^4 + 10xz^5 + 9z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 11x^4 + 18x^3 + 37x^2 + 40x + 36$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([9, 10, 9, 4, 2]), R([0, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![9, 10, 9, 4, 2], R![0, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([36, 40, 37, 18, 11, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(1145\) = \( 5 \cdot 229 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-143125\) = \( - 5^{4} \cdot 229 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-10008\) =  \( - 2^{3} \cdot 3^{2} \cdot 139 \)
\( I_4 \)  = \(764388\) =  \( 2^{2} \cdot 3^{2} \cdot 17 \cdot 1249 \)
\( I_6 \)  = \(-2318851224\) =  \( - 2^{3} \cdot 3^{2} \cdot 293 \cdot 109919 \)
\( I_{10} \)  = \(-586240000\) =  \( - 2^{12} \cdot 5^{4} \cdot 229 \)
\( J_2 \)  = \(-1251\) =  \( - 3^{2} \cdot 139 \)
\( J_4 \)  = \(57246\) =  \( 2 \cdot 3 \cdot 7 \cdot 29 \cdot 47 \)
\( J_6 \)  = \(-3273124\) =  \( - 2^{2} \cdot 818281 \)
\( J_8 \)  = \(204393402\) =  \( 2 \cdot 3^{3} \cdot 37 \cdot 102299 \)
\( J_{10} \)  = \(-143125\) =  \( - 5^{4} \cdot 229 \)
\( g_1 \)  = \(3063984390631251/143125\)
\( g_2 \)  = \(112077149104746/143125\)
\( g_3 \)  = \(5122442333124/143125\)

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

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

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : -2 : 1),\, (0 : -3 : 1),\, (0 : 3 : 1),\, (-1 : 3 : 1)\)

magma: [C![-1,-2,1],C![-1,3,1],C![0,-3,1],C![0,3,1],C![1,-1,0],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: \(\Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + xz + 9z^2\) \(=\) \(0,\) \(4y\) \(=\) \(7xz^2 + 9z^3\) \(0.026503\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(3xz^2 + 4z^3\) \(0\) \(2\)

2-torsion field: 6.4.3356224.3

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.026503 \)
Real period: \( 7.396287 \)
Tamagawa product: \( 4 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.196027 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(5\) \(4\) \(1\) \(4\) \(( 1 - T )( 1 + 3 T + 5 T^{2} )\)
\(229\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 229 T^{2} )\)

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 \(\Q\).