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Properties

Label 61099.a.61099.1
Conductor $61099$
Discriminant $-61099$
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

$y^2 + (x^3 + 1)y = -3x^4 + 3x^3 + 2x^2 - 2x$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -3x^4z^2 + 3x^3z^3 + 2x^2z^4 - 2xz^5$ (dehomogenize, simplify)
$y^2 = x^6 - 12x^4 + 14x^3 + 8x^2 - 8x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 2, 3, -3]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 2, 3, -3], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, -8, 8, 14, -12, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(61099\) \(=\) \( 61099 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-61099\) \(=\) \( -61099 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1236\) \(=\)  \( 2^{2} \cdot 3 \cdot 103 \)
\( I_4 \)  \(=\) \(54105\) \(=\)  \( 3 \cdot 5 \cdot 3607 \)
\( I_6 \)  \(=\) \(18692061\) \(=\)  \( 3 \cdot 17 \cdot 366511 \)
\( I_{10} \)  \(=\) \(-7820672\) \(=\)  \( - 2^{7} \cdot 61099 \)
\( J_2 \)  \(=\) \(309\) \(=\)  \( 3 \cdot 103 \)
\( J_4 \)  \(=\) \(1724\) \(=\)  \( 2^{2} \cdot 431 \)
\( J_6 \)  \(=\) \(2184\) \(=\)  \( 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
\( J_8 \)  \(=\) \(-574330\) \(=\)  \( - 2 \cdot 5 \cdot 79 \cdot 727 \)
\( J_{10} \)  \(=\) \(-61099\) \(=\)  \( -61099 \)
\( g_1 \)  \(=\) \(-2817036000549/61099\)
\( g_2 \)  \(=\) \(-50864256396/61099\)
\( g_3 \)  \(=\) \(-208530504/61099\)

  (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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -2 : 1)\)
\((1 : -4 : 2)\) \((2 : -4 : 1)\) \((-1 : 5 : 2)\) \((1 : -5 : 2)\) \((2 : -5 : 1)\) \((-1 : -12 : 2)\)
\((-4 : 23 : 1)\) \((-4 : 40 : 1)\) \((12 : -735 : 7)\) \((12 : -1336 : 7)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -2 : 1)\)
\((1 : -4 : 2)\) \((2 : -4 : 1)\) \((-1 : 5 : 2)\) \((1 : -5 : 2)\) \((2 : -5 : 1)\) \((-1 : -12 : 2)\)
\((-4 : 23 : 1)\) \((-4 : 40 : 1)\) \((12 : -735 : 7)\) \((12 : -1336 : 7)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : -1 : 2)\) \((1 : 1 : 2)\)
\((1 : -2 : 1)\) \((1 : 2 : 1)\) \((2 : -1 : 1)\) \((2 : 1 : 1)\) \((-1 : -17 : 2)\) \((-1 : 17 : 2)\)
\((-4 : -17 : 1)\) \((-4 : 17 : 1)\) \((12 : -601 : 7)\) \((12 : 601 : 7)\)

magma: [C![-4,23,1],C![-4,40,1],C![-1,-12,2],C![-1,5,2],C![0,-1,1],C![0,0,1],C![1,-5,2],C![1,-4,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-5,1],C![2,-4,1],C![12,-1336,7],C![12,-735,7]]; // minimal model
 
magma: [C![-4,-17,1],C![-4,17,1],C![-1,-17,2],C![-1,17,2],C![0,-1,1],C![0,1,1],C![1,-1,2],C![1,1,2],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1],C![2,-1,1],C![2,1,1],C![12,-601,7],C![12,601,7]]; // 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
\((0 : -1 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0.577534\) \(\infty\)
\((0 : -1 : 1) + (2 : -5 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0.242037\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.313064\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0.577534\) \(\infty\)
\((0 : -1 : 1) + (2 : -5 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0.242037\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.313064\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) + (1 : 2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 2xz^2 - z^3\) \(0.577534\) \(\infty\)
\((0 : -1 : 1) + (2 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 4xz^2 - z^3\) \(0.242037\) \(\infty\)
\((0 : -1 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.313064\) \(\infty\)

2-torsion field: 6.4.3910336.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.032785 \)
Real period: \( 21.85533 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.716543 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(61099\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 371 T + 61099 T^{2} )\)

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);