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Properties

Label 776117.a.776117.1
Conductor $776117$
Discriminant $-776117$
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

$y^2 + (x^3 + x^2 + x)y = 2x^4 + 2x^3 + 3x^2 + 2x + 1$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2)y = 2x^4z^2 + 2x^3z^3 + 3x^2z^4 + 2xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 11x^4 + 10x^3 + 13x^2 + 8x + 4$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 2, 3, 2, 2]), R([0, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 2, 3, 2, 2], R![0, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([4, 8, 13, 10, 11, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(776117\) \(=\) \( 776117 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-776117\) \(=\) \( -776117 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1004\) \(=\)  \( 2^{2} \cdot 251 \)
\( I_4 \)  \(=\) \(23257\) \(=\)  \( 13 \cdot 1789 \)
\( I_6 \)  \(=\) \(5964515\) \(=\)  \( 5 \cdot 1192903 \)
\( I_{10} \)  \(=\) \(99342976\) \(=\)  \( 2^{7} \cdot 776117 \)
\( J_2 \)  \(=\) \(251\) \(=\)  \( 251 \)
\( J_4 \)  \(=\) \(1656\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 23 \)
\( J_6 \)  \(=\) \(21328\) \(=\)  \( 2^{4} \cdot 31 \cdot 43 \)
\( J_8 \)  \(=\) \(652748\) \(=\)  \( 2^{2} \cdot 53 \cdot 3079 \)
\( J_{10} \)  \(=\) \(776117\) \(=\)  \( 776117 \)
\( g_1 \)  \(=\) \(996250626251/776117\)
\( g_2 \)  \(=\) \(26186743656/776117\)
\( g_3 \)  \(=\) \(1343685328/776117\)

  (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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 2 : 1)\)
\((1 : 2 : 1)\) \((1 : -5 : 1)\) \((-1 : -5 : 2)\) \((-3 : -5 : 1)\) \((-1 : 8 : 2)\) \((-1 : -18 : 3)\)
\((-1 : 25 : 3)\) \((-3 : 26 : 1)\) \((5 : 37 : 2)\) \((-5 : -58 : 3)\) \((-5 : 153 : 3)\) \((5 : -232 : 2)\)
\((-39 : -873800 : 110)\) \((-39 : 1237709 : 110)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 2 : 1)\)
\((1 : 2 : 1)\) \((1 : -5 : 1)\) \((-1 : -5 : 2)\) \((-3 : -5 : 1)\) \((-1 : 8 : 2)\) \((-1 : -18 : 3)\)
\((-1 : 25 : 3)\) \((-3 : 26 : 1)\) \((5 : 37 : 2)\) \((-5 : -58 : 3)\) \((-5 : 153 : 3)\) \((5 : -232 : 2)\)
\((-39 : -873800 : 110)\) \((-39 : 1237709 : 110)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -2 : 1)\) \((0 : 2 : 1)\) \((-1 : -3 : 1)\) \((-1 : 3 : 1)\)
\((1 : -7 : 1)\) \((1 : 7 : 1)\) \((-1 : -13 : 2)\) \((-1 : 13 : 2)\) \((-3 : -31 : 1)\) \((-3 : 31 : 1)\)
\((-1 : -43 : 3)\) \((-1 : 43 : 3)\) \((-5 : -211 : 3)\) \((-5 : 211 : 3)\) \((5 : -269 : 2)\) \((5 : 269 : 2)\)
\((-39 : -2111509 : 110)\) \((-39 : 2111509 : 110)\)

magma: [C![-39,-873800,110],C![-39,1237709,110],C![-5,-58,3],C![-5,153,3],C![-3,-5,1],C![-3,26,1],C![-1,-18,3],C![-1,-5,2],C![-1,-1,1],C![-1,2,1],C![-1,8,2],C![-1,25,3],C![0,-1,1],C![0,1,1],C![1,-5,1],C![1,-1,0],C![1,0,0],C![1,2,1],C![5,-232,2],C![5,37,2]]; // minimal model
 
magma: [C![-39,-2111509,110],C![-39,2111509,110],C![-5,-211,3],C![-5,211,3],C![-3,-31,1],C![-3,31,1],C![-1,-43,3],C![-1,-13,2],C![-1,-3,1],C![-1,3,1],C![-1,13,2],C![-1,43,3],C![0,-2,1],C![0,2,1],C![1,-7,1],C![1,-1,0],C![1,1,0],C![1,7,1],C![5,-269,2],C![5,269,2]]; // 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
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + z^3\) \(0.828606\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.759844\) \(\infty\)
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.827291\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.628713\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + z^3\) \(0.828606\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.759844\) \(\infty\)
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.827291\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.628713\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + 3xz^2 + 2z^3\) \(0.828606\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + xz^2\) \(0.759844\) \(\infty\)
\((-1 : -3 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + xz^2 - 2z^3\) \(0.827291\) \(\infty\)
\((0 : -2 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + x^2z + xz^2 - 2z^3\) \(0.628713\) \(\infty\)

2-torsion field: 6.0.49671488.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(4\)   (upper bound)
Mordell-Weil rank: \(4\)
2-Selmer rank:\(4\)
Regulator: \( 0.185467 \)
Real period: \( 10.28881 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.908246 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(776117\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 986 T + 776117 T^{2} )\)

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);