Properties

Label 93162.a.558972.1
Conductor $93162$
Discriminant $-558972$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = -2x^4 + x^3 + 2x^2 - 3x + 2$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -2x^4z^2 + x^3z^3 + 2x^2z^4 - 3xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 8x^4 + 6x^3 + 8x^2 - 12x + 9$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -3, 2, 1, -2]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -3, 2, 1, -2], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([9, -12, 8, 6, -8, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(93162\) \(=\) \( 2 \cdot 3 \cdot 15527 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-558972\) \(=\) \( - 2^{2} \cdot 3^{2} \cdot 15527 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(460\) \(=\)  \( 2^{2} \cdot 5 \cdot 23 \)
\( I_4 \)  \(=\) \(39913\) \(=\)  \( 167 \cdot 239 \)
\( I_6 \)  \(=\) \(11017259\) \(=\)  \( 11 \cdot 1001569 \)
\( I_{10} \)  \(=\) \(71548416\) \(=\)  \( 2^{9} \cdot 3^{2} \cdot 15527 \)
\( J_2 \)  \(=\) \(115\) \(=\)  \( 5 \cdot 23 \)
\( J_4 \)  \(=\) \(-1112\) \(=\)  \( - 2^{3} \cdot 139 \)
\( J_6 \)  \(=\) \(-96372\) \(=\)  \( - 2^{2} \cdot 3^{2} \cdot 2677 \)
\( J_8 \)  \(=\) \(-3079831\) \(=\)  \( -3079831 \)
\( J_{10} \)  \(=\) \(558972\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 15527 \)
\( g_1 \)  \(=\) \(20113571875/558972\)
\( g_2 \)  \(=\) \(-422803250/139743\)
\( g_3 \)  \(=\) \(-35403325/15527\)

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

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\) \((0 : -2 : 1)\) \((-1 : -2 : 1)\)
\((-1 : 2 : 1)\) \((1 : -2 : 1)\) \((2 : -4 : 1)\) \((2 : -5 : 1)\) \((-3 : 10 : 1)\) \((-4 : 10 : 1)\)
\((-3 : 16 : 1)\) \((3 : -16 : 2)\) \((3 : -19 : 2)\) \((5 : -37 : 2)\) \((-4 : 53 : 1)\) \((5 : -96 : 2)\)
\((15 : 12256 : 28)\) \((15 : -37583 : 28)\) \((-43 : 179982 : 45)\) \((-43 : -191600 : 45)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\) \((0 : -2 : 1)\) \((-1 : -2 : 1)\)
\((-1 : 2 : 1)\) \((1 : -2 : 1)\) \((2 : -4 : 1)\) \((2 : -5 : 1)\) \((-3 : 10 : 1)\) \((-4 : 10 : 1)\)
\((-3 : 16 : 1)\) \((3 : -16 : 2)\) \((3 : -19 : 2)\) \((5 : -37 : 2)\) \((-4 : 53 : 1)\) \((5 : -96 : 2)\)
\((15 : 12256 : 28)\) \((15 : -37583 : 28)\) \((-43 : 179982 : 45)\) \((-43 : -191600 : 45)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((1 : -2 : 1)\) \((1 : 2 : 1)\) \((2 : -1 : 1)\) \((2 : 1 : 1)\)
\((0 : -3 : 1)\) \((0 : 3 : 1)\) \((3 : -3 : 2)\) \((3 : 3 : 2)\) \((-1 : -4 : 1)\) \((-1 : 4 : 1)\)
\((-3 : -6 : 1)\) \((-3 : 6 : 1)\) \((-4 : -43 : 1)\) \((-4 : 43 : 1)\) \((5 : -59 : 2)\) \((5 : 59 : 2)\)
\((15 : -49839 : 28)\) \((15 : 49839 : 28)\) \((-43 : -371582 : 45)\) \((-43 : 371582 : 45)\)

magma: [C![-43,-191600,45],C![-43,179982,45],C![-4,10,1],C![-4,53,1],C![-3,10,1],C![-3,16,1],C![-1,-2,1],C![-1,2,1],C![0,-2,1],C![0,1,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-5,1],C![2,-4,1],C![3,-19,2],C![3,-16,2],C![5,-96,2],C![5,-37,2],C![15,-37583,28],C![15,12256,28]]; // minimal model
 
magma: [C![-43,-371582,45],C![-43,371582,45],C![-4,-43,1],C![-4,43,1],C![-3,-6,1],C![-3,6,1],C![-1,-4,1],C![-1,4,1],C![0,-3,1],C![0,3,1],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1],C![2,-1,1],C![2,1,1],C![3,-3,2],C![3,3,2],C![5,-59,2],C![5,59,2],C![15,-49839,28],C![15,49839,28]]; // simplified model
 

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: \(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.396197\) \(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.202076\) \(\infty\)
\((0 : 1 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + z^3\) \(0.202984\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.396197\) \(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.202076\) \(\infty\)
\((0 : 1 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 + z^3\) \(0.202984\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + z^3\) \(0.396197\) \(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3z^3\) \(0.202076\) \(\infty\)
\((0 : 3 : 1) + (1 : 2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 2xz^2 + 3z^3\) \(0.202984\) \(\infty\)

2-torsion field: 6.4.993728.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.014247 \)
Real period: \( 16.00424 \)
Tamagawa product: \( 4 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.912113 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(3\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(15527\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 99 T + 15527 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

magma: HeuristicDecompositionFactors(C);
 

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 \(\Q\).

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);