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Properties

Label 87767.a.87767.1
Conductor $87767$
Discriminant $87767$
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

$y^2 + (x^3 + x + 1)y = -x^4 + x^3 - 3x^2 + x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 + x^3z^3 - 3x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = x^6 - 2x^4 + 6x^3 - 11x^2 + 6x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -3, 1, -1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -3, 1, -1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 6, -11, 6, -2, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(87767\) \(=\) \( 87767 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(87767\) \(=\) \( 87767 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(188\) \(=\)  \( 2^{2} \cdot 47 \)
\( I_4 \)  \(=\) \(-3047\) \(=\)  \( - 11 \cdot 277 \)
\( I_6 \)  \(=\) \(45203\) \(=\)  \( 17 \cdot 2659 \)
\( I_{10} \)  \(=\) \(-11234176\) \(=\)  \( - 2^{7} \cdot 87767 \)
\( J_2 \)  \(=\) \(47\) \(=\)  \( 47 \)
\( J_4 \)  \(=\) \(219\) \(=\)  \( 3 \cdot 73 \)
\( J_6 \)  \(=\) \(-2045\) \(=\)  \( - 5 \cdot 409 \)
\( J_8 \)  \(=\) \(-36019\) \(=\)  \( - 181 \cdot 199 \)
\( J_{10} \)  \(=\) \(-87767\) \(=\)  \( -87767 \)
\( g_1 \)  \(=\) \(-229345007/87767\)
\( g_2 \)  \(=\) \(-22737237/87767\)
\( g_3 \)  \(=\) \(4517405/87767\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : -1 : 1)\) \((1 : -2 : 1)\)
\((1 : -1 : 2)\) \((2 : -2 : 1)\) \((-3 : 6 : 1)\) \((2 : -9 : 1)\) \((1 : -12 : 2)\) \((-3 : 23 : 1)\)
\((10 : -82 : 3)\) \((7 : -279 : 6)\) \((7 : -532 : 6)\) \((10 : -1035 : 3)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : -1 : 1)\) \((1 : -2 : 1)\)
\((1 : -1 : 2)\) \((2 : -2 : 1)\) \((-3 : 6 : 1)\) \((2 : -9 : 1)\) \((1 : -12 : 2)\) \((-3 : 23 : 1)\)
\((10 : -82 : 3)\) \((7 : -279 : 6)\) \((7 : -532 : 6)\) \((10 : -1035 : 3)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : -1 : 1)\) \((1 : 1 : 1)\)
\((2 : -7 : 1)\) \((2 : 7 : 1)\) \((1 : -11 : 2)\) \((1 : 11 : 2)\) \((-3 : -17 : 1)\) \((-3 : 17 : 1)\)
\((7 : -253 : 6)\) \((7 : 253 : 6)\) \((10 : -953 : 3)\) \((10 : 953 : 3)\)

magma: [C![-3,6,1],C![-3,23,1],C![0,-1,1],C![0,0,1],C![1,-12,2],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,-1,2],C![1,0,0],C![2,-9,1],C![2,-2,1],C![7,-532,6],C![7,-279,6],C![10,-1035,3],C![10,-82,3]]; // minimal model
 
magma: [C![-3,-17,1],C![-3,17,1],C![0,-1,1],C![0,1,1],C![1,-11,2],C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,11,2],C![1,1,0],C![2,-7,1],C![2,7,1],C![7,-253,6],C![7,253,6],C![10,-953,3],C![10,953,3]]; // 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
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.504991\) \(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.442956\) \(\infty\)
\((1 : -2 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.333441\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.504991\) \(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.442956\) \(\infty\)
\((1 : -2 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.333441\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 - z^3\) \(0.504991\) \(\infty\)
\((1 : -1 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 - 3z^3\) \(0.442956\) \(\infty\)
\((1 : -1 : 1) - (1 : 1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + xz^2 - z^3\) \(0.333441\) \(\infty\)

2-torsion field: 6.2.5617088.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.045308 \)
Real period: \( 14.46119 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.655214 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(87767\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 498 T + 87767 T^{2} )\)

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);