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Properties

Label 766561.b.766561.1
Conductor $766561$
Discriminant $766561$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z \oplus \Z\)
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

$y^2 + (x^3 + x + 1)y = -x^5 + x^3 - 6x + 6$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^5z + x^3z^3 - 6xz^5 + 6z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 4x^5 + 2x^4 + 6x^3 + x^2 - 22x + 25$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([6, -6, 0, 1, 0, -1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![6, -6, 0, 1, 0, -1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([25, -22, 1, 6, 2, -4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(766561\) \(=\) \( 43 \cdot 17827 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(766561\) \(=\) \( 43 \cdot 17827 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1148\) \(=\)  \( 2^{2} \cdot 7 \cdot 41 \)
\( I_4 \)  \(=\) \(63145\) \(=\)  \( 5 \cdot 73 \cdot 173 \)
\( I_6 \)  \(=\) \(26374427\) \(=\)  \( 29 \cdot 909463 \)
\( I_{10} \)  \(=\) \(-98119808\) \(=\)  \( - 2^{7} \cdot 43 \cdot 17827 \)
\( J_2 \)  \(=\) \(287\) \(=\)  \( 7 \cdot 41 \)
\( J_4 \)  \(=\) \(801\) \(=\)  \( 3^{2} \cdot 89 \)
\( J_6 \)  \(=\) \(-101837\) \(=\)  \( -101837 \)
\( J_8 \)  \(=\) \(-7467205\) \(=\)  \( - 5 \cdot 1493441 \)
\( J_{10} \)  \(=\) \(-766561\) \(=\)  \( - 43 \cdot 17827 \)
\( g_1 \)  \(=\) \(-1947195170207/766561\)
\( g_2 \)  \(=\) \(-18935562303/766561\)
\( g_3 \)  \(=\) \(8388211853/766561\)

  (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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((1 : 0 : 1)\) \((0 : 2 : 1)\) \((0 : -3 : 1)\) \((-1 : -3 : 1)\)
\((1 : -3 : 1)\) \((-1 : 4 : 1)\) \((2 : -5 : 1)\) \((2 : -6 : 1)\) \((1 : 9 : 2)\) \((3 : -12 : 1)\)
\((3 : -14 : 2)\) \((3 : -19 : 1)\) \((1 : -22 : 2)\) \((3 : -33 : 2)\) \((-4 : -77 : 3)\) \((-4 : 150 : 3)\)
\((8 : -266 : 3)\) \((8 : -345 : 3)\) \((-13 : -401 : 3)\) \((-13 : 2688 : 3)\) \((3869 : -22201537065 : 1147)\) \((3869 : -42313241588 : 1147)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((1 : 0 : 1)\) \((0 : 2 : 1)\) \((0 : -3 : 1)\) \((-1 : -3 : 1)\)
\((1 : -3 : 1)\) \((-1 : 4 : 1)\) \((2 : -5 : 1)\) \((2 : -6 : 1)\) \((1 : 9 : 2)\) \((3 : -12 : 1)\)
\((3 : -14 : 2)\) \((3 : -19 : 1)\) \((1 : -22 : 2)\) \((3 : -33 : 2)\) \((-4 : -77 : 3)\) \((-4 : 150 : 3)\)
\((8 : -266 : 3)\) \((8 : -345 : 3)\) \((-13 : -401 : 3)\) \((-13 : 2688 : 3)\) \((3869 : -22201537065 : 1147)\) \((3869 : -42313241588 : 1147)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((2 : -1 : 1)\) \((2 : 1 : 1)\) \((1 : -3 : 1)\) \((1 : 3 : 1)\)
\((0 : -5 : 1)\) \((0 : 5 : 1)\) \((-1 : -7 : 1)\) \((-1 : 7 : 1)\) \((3 : -7 : 1)\) \((3 : 7 : 1)\)
\((3 : -19 : 2)\) \((3 : 19 : 2)\) \((1 : -31 : 2)\) \((1 : 31 : 2)\) \((8 : -79 : 3)\) \((8 : 79 : 3)\)
\((-4 : -227 : 3)\) \((-4 : 227 : 3)\) \((-13 : -3089 : 3)\) \((-13 : 3089 : 3)\) \((3869 : -20111704523 : 1147)\) \((3869 : 20111704523 : 1147)\)

magma: [C![-13,-401,3],C![-13,2688,3],C![-4,-77,3],C![-4,150,3],C![-1,-3,1],C![-1,4,1],C![0,-3,1],C![0,2,1],C![1,-22,2],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![1,9,2],C![2,-6,1],C![2,-5,1],C![3,-33,2],C![3,-19,1],C![3,-14,2],C![3,-12,1],C![8,-345,3],C![8,-266,3],C![3869,-42313241588,1147],C![3869,-22201537065,1147]]; // minimal model
 
magma: [C![-13,-3089,3],C![-13,3089,3],C![-4,-227,3],C![-4,227,3],C![-1,-7,1],C![-1,7,1],C![0,-5,1],C![0,5,1],C![1,-31,2],C![1,-3,1],C![1,-1,0],C![1,1,0],C![1,3,1],C![1,31,2],C![2,-1,1],C![2,1,1],C![3,-19,2],C![3,-7,1],C![3,19,2],C![3,7,1],C![8,-79,3],C![8,79,3],C![3869,-20111704523,1147],C![3869,20111704523,1147]]; // 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: \(\Z \oplus \Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -3 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-3z^3\) \(0.910226\) \(\infty\)
\((2 : -6 : 1) - (1 : -1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-6z^3\) \(0.395918\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-4xz^2 + z^3\) \(0.945217\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(-x^2z\) \(0.401748\) \(\infty\)
Generator $D_0$ Height Order
\((1 : -3 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-3z^3\) \(0.910226\) \(\infty\)
\((2 : -6 : 1) - (1 : -1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-6z^3\) \(0.395918\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-4xz^2 + z^3\) \(0.945217\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(-x^2z\) \(0.401748\) \(\infty\)
Generator $D_0$ Height Order
\((1 : -3 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 - 5z^3\) \(0.910226\) \(\infty\)
\((2 : -1 : 1) - (1 : -1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 - 11z^3\) \(0.395918\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 7xz^2 + 3z^3\) \(0.945217\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 2x^2z + xz^2 + z^3\) \(0.401748\) \(\infty\)

2-torsion field: 6.2.49059904.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(4\)   (upper bound)
Mordell-Weil rank: \(4\)
2-Selmer rank:\(4\)
Regulator: \( 0.101613 \)
Real period: \( 14.96038 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.520181 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(43\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 43 T^{2} )\)
\(17827\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 244 T + 17827 T^{2} )\)

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);