Properties

Label 1777.a.1777.1
Conductor $1777$
Discriminant $1777$
Mordell-Weil group \(\Z\)
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

$y^2 + (x^3 + x + 1)y = -3x^4 + 7x^2 - x - 8$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -3x^4z^2 + 7x^2z^4 - xz^5 - 8z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 10x^4 + 2x^3 + 29x^2 - 2x - 31$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-8, -1, 7, 0, -3]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-8, -1, 7, 0, -3], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-31, -2, 29, 2, -10, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1777\) \(=\) \( 1777 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(1777\) \(=\) \( 1777 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(6052\) \(=\)  \( 2^{2} \cdot 17 \cdot 89 \)
\( I_4 \)  \(=\) \(-1391\) \(=\)  \( - 13 \cdot 107 \)
\( I_6 \)  \(=\) \(-2704039\) \(=\)  \( - 13 \cdot 208003 \)
\( I_{10} \)  \(=\) \(227456\) \(=\)  \( 2^{7} \cdot 1777 \)
\( J_2 \)  \(=\) \(1513\) \(=\)  \( 17 \cdot 89 \)
\( J_4 \)  \(=\) \(95440\) \(=\)  \( 2^{4} \cdot 5 \cdot 1193 \)
\( J_6 \)  \(=\) \(8030588\) \(=\)  \( 2^{2} \cdot 23 \cdot 41 \cdot 2129 \)
\( J_8 \)  \(=\) \(760371511\) \(=\)  \( 6353 \cdot 119687 \)
\( J_{10} \)  \(=\) \(1777\) \(=\)  \( 1777 \)
\( g_1 \)  \(=\) \(7928565897078793/1777\)
\( g_2 \)  \(=\) \(330557651801680/1777\)
\( g_3 \)  \(=\) \(18383373101372/1777\)

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

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (2 : -5 : 1),\, (2 : -6 : 1),\, (-4 : 12 : 1),\, (-4 : 55 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (2 : -5 : 1),\, (2 : -6 : 1),\, (-4 : 12 : 1),\, (-4 : 55 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (2 : -1 : 1),\, (2 : 1 : 1),\, (-4 : -43 : 1),\, (-4 : 43 : 1)\)

magma: [C![-4,12,1],C![-4,55,1],C![1,-1,0],C![1,0,0],C![2,-6,1],C![2,-5,1]]; // minimal model
 
magma: [C![-4,-43,1],C![-4,43,1],C![1,-1,0],C![1,1,0],C![2,-1,1],C![2,1,1]]; // 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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((2 : -6 : 1) - (1 : 0 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + 2z^3\) \(0.039015\) \(\infty\)
Generator $D_0$ Height Order
\((2 : -6 : 1) - (1 : 0 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + 2z^3\) \(0.039015\) \(\infty\)
Generator $D_0$ Height Order
\((2 : -1 : 1) - (1 : 1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + xz^2 + 5z^3\) \(0.039015\) \(\infty\)

2-torsion field: 6.2.113728.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.039015 \)
Real period: \( 7.694781 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.300215 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(1777\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 37 T + 1777 T^{2} )\)

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);