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Properties

Label 61127.a.61127.1
Conductor $61127$
Discriminant $-61127$
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

$y^2 + (x^3 + x + 1)y = 2x^5 + 3x^4 - x^2$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = 2x^5z + 3x^4z^2 - x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 + 8x^5 + 14x^4 + 2x^3 - 3x^2 + 2x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -1, 0, 3, 2]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -1, 0, 3, 2], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, -3, 2, 14, 8, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(61127\) \(=\) \( 11 \cdot 5557 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-61127\) \(=\) \( - 11 \cdot 5557 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(548\) \(=\)  \( 2^{2} \cdot 137 \)
\( I_4 \)  \(=\) \(10489\) \(=\)  \( 17 \cdot 617 \)
\( I_6 \)  \(=\) \(1460957\) \(=\)  \( 1460957 \)
\( I_{10} \)  \(=\) \(-7824256\) \(=\)  \( - 2^{7} \cdot 11 \cdot 5557 \)
\( J_2 \)  \(=\) \(137\) \(=\)  \( 137 \)
\( J_4 \)  \(=\) \(345\) \(=\)  \( 3 \cdot 5 \cdot 23 \)
\( J_6 \)  \(=\) \(2293\) \(=\)  \( 2293 \)
\( J_8 \)  \(=\) \(48779\) \(=\)  \( 48779 \)
\( J_{10} \)  \(=\) \(-61127\) \(=\)  \( - 11 \cdot 5557 \)
\( g_1 \)  \(=\) \(-48261724457/61127\)
\( g_2 \)  \(=\) \(-887116785/61127\)
\( g_3 \)  \(=\) \(-43037317/61127\)

  (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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((1 : 0 : 2)\) \((1 : -4 : 1)\) \((-2 : 4 : 1)\) \((-2 : 5 : 1)\) \((-1 : -5 : 3)\)
\((-1 : -12 : 3)\) \((1 : -13 : 2)\) \((3 : 25 : 2)\) \((3 : -72 : 2)\) \((-10 : 175 : 7)\) \((-10 : 972 : 7)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((1 : 0 : 2)\) \((1 : -4 : 1)\) \((-2 : 4 : 1)\) \((-2 : 5 : 1)\) \((-1 : -5 : 3)\)
\((-1 : -12 : 3)\) \((1 : -13 : 2)\) \((3 : 25 : 2)\) \((3 : -72 : 2)\) \((-10 : 175 : 7)\) \((-10 : 972 : 7)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((-2 : -1 : 1)\) \((-2 : 1 : 1)\) \((1 : -5 : 1)\) \((1 : 5 : 1)\) \((-1 : -7 : 3)\) \((-1 : 7 : 3)\)
\((1 : -13 : 2)\) \((1 : 13 : 2)\) \((3 : -97 : 2)\) \((3 : 97 : 2)\) \((-10 : -797 : 7)\) \((-10 : 797 : 7)\)

magma: [C![-10,175,7],C![-10,972,7],C![-2,4,1],C![-2,5,1],C![-1,-12,3],C![-1,-5,3],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-13,2],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,0,2],C![1,1,1],C![3,-72,2],C![3,25,2]]; // minimal model
 
magma: [C![-10,-797,7],C![-10,797,7],C![-2,-1,1],C![-2,1,1],C![-1,-7,3],C![-1,7,3],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-13,2],C![1,-5,1],C![1,-1,0],C![1,1,0],C![1,13,2],C![1,5,1],C![3,-97,2],C![3,97,2]]; // 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: \(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.334365\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.287560\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.368264\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.334365\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.287560\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.368264\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 + z^3\) \(0.334365\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 - z^3\) \(0.287560\) \(\infty\)
\((-1 : -1 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + xz^2 - z^3\) \(0.368264\) \(\infty\)

2-torsion field: 6.4.3912128.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.029475 \)
Real period: \( 19.61438 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.578137 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 6 T + 11 T^{2} )\)
\(5557\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 147 T + 5557 T^{2} )\)

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);