Properties

Label 18080.c.723200.1
Conductor 18080
Discriminant -723200
Mordell-Weil group \(\Z \times \Z \times \Z/{2}\Z\)
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

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -2, 1, 1, 0, -2, 1], R![0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -2, 1, 1, 0, -2, 1]), R([0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([4, -8, 5, 4, 0, -8, 4]))
 

Invariants

Conductor: \( N \)  =  \(18080\) = \( 2^{5} \cdot 5 \cdot 113 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-723200\) = \( - 2^{8} \cdot 5^{2} \cdot 113 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-1184\) =  \( - 2^{5} \cdot 37 \)
\( I_4 \)  = \(186688\) =  \( 2^{6} \cdot 2917 \)
\( I_6 \)  = \(-76176896\) =  \( - 2^{9} \cdot 148783 \)
\( I_{10} \)  = \(-2962227200\) =  \( - 2^{20} \cdot 5^{2} \cdot 113 \)
\( J_2 \)  = \(-148\) =  \( - 2^{2} \cdot 37 \)
\( J_4 \)  = \(-1032\) =  \( - 2^{3} \cdot 3 \cdot 43 \)
\( J_6 \)  = \(44800\) =  \( 2^{8} \cdot 5^{2} \cdot 7 \)
\( J_8 \)  = \(-1923856\) =  \( - 2^{4} \cdot 11 \cdot 17 \cdot 643 \)
\( J_{10} \)  = \(-723200\) =  \( - 2^{8} \cdot 5^{2} \cdot 113 \)
\( g_1 \)  = \(277375828/2825\)
\( g_2 \)  = \(-13068474/2825\)
\( g_3 \)  = \(-153328/113\)

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
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\)
\((-1 : -2 : 1)\) \((-1 : 3 : 1)\) \((1 : 3 : 2)\) \((3 : -1 : 2)\) \((1 : -7 : 2)\) \((3 : -11 : 2)\)
\((7 : -41 : 6)\) \((7 : -211 : 6)\)

magma: [C![-1,-2,1],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,-7,2],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![1,3,2],C![3,-11,2],C![3,-1,2],C![7,-211,6],C![7,-41,6]];
 

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: \(\Z \times \Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.557085\) \(\infty\)
\((0 : -1 : 1) + (3 : -11 : 2) - (1 : -1 : 0) - (1 : 1 : 0)\) \((-2x + 3z) x\) \(=\) \(0,\) \(4y\) \(=\) \(-xz^2 - 4z^3\) \(0.039896\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(2x^2 - 2xz + z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)

2-torsion field: 4.0.1808.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(3\)
Regulator: \( 0.022196 \)
Real period: \( 14.76743 \)
Tamagawa product: \( 8 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.655575 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(8\) \(5\) \(4\) \(1 + T\)
\(5\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(113\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 14 T + 113 T^{2} )\)

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 \(\Q\).