Properties

Label 79337.a.79337.1
Conductor $79337$
Discriminant $79337$
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

$y^2 + (x^3 + x + 1)y = -x^4 + x^3 - x^2 - 3x + 2$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 + x^3z^3 - x^2z^4 - 3xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 2x^4 + 6x^3 - 3x^2 - 10x + 9$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -3, -1, 1, -1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -3, -1, 1, -1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([9, -10, -3, 6, -2, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(79337\) \(=\) \( 79337 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(79337\) \(=\) \( 79337 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1020\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 17 \)
\( I_4 \)  \(=\) \(28713\) \(=\)  \( 3 \cdot 17 \cdot 563 \)
\( I_6 \)  \(=\) \(10231131\) \(=\)  \( 3 \cdot 59 \cdot 57803 \)
\( I_{10} \)  \(=\) \(-10155136\) \(=\)  \( - 2^{7} \cdot 79337 \)
\( J_2 \)  \(=\) \(255\) \(=\)  \( 3 \cdot 5 \cdot 17 \)
\( J_4 \)  \(=\) \(1513\) \(=\)  \( 17 \cdot 89 \)
\( J_6 \)  \(=\) \(-18973\) \(=\)  \( -18973 \)
\( J_8 \)  \(=\) \(-1781821\) \(=\)  \( - 17 \cdot 281 \cdot 373 \)
\( J_{10} \)  \(=\) \(-79337\) \(=\)  \( -79337 \)
\( g_1 \)  \(=\) \(-1078203909375/79337\)
\( g_2 \)  \(=\) \(-25087620375/79337\)
\( g_3 \)  \(=\) \(1233719325/79337\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((1 : -1 : 1)\) \((0 : -2 : 1)\)
\((-1 : 2 : 1)\) \((1 : -2 : 1)\) \((-2 : 4 : 1)\) \((-2 : 5 : 1)\) \((7 : -23 : 2)\) \((4 : -37 : 3)\)
\((4 : -90 : 3)\) \((7 : -356 : 2)\) \((11 : -1206 : 15)\) \((11 : -5975 : 15)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((1 : -1 : 1)\) \((0 : -2 : 1)\)
\((-1 : 2 : 1)\) \((1 : -2 : 1)\) \((-2 : 4 : 1)\) \((-2 : 5 : 1)\) \((7 : -23 : 2)\) \((4 : -37 : 3)\)
\((4 : -90 : 3)\) \((7 : -356 : 2)\) \((11 : -1206 : 15)\) \((11 : -5975 : 15)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((1 : -1 : 1)\) \((1 : 1 : 1)\) \((-2 : -1 : 1)\) \((-2 : 1 : 1)\)
\((0 : -3 : 1)\) \((0 : 3 : 1)\) \((-1 : -3 : 1)\) \((-1 : 3 : 1)\) \((4 : -53 : 3)\) \((4 : 53 : 3)\)
\((7 : -333 : 2)\) \((7 : 333 : 2)\) \((11 : -4769 : 15)\) \((11 : 4769 : 15)\)

magma: [C![-2,4,1],C![-2,5,1],C![-1,-1,1],C![-1,2,1],C![0,-2,1],C![0,1,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![4,-90,3],C![4,-37,3],C![7,-356,2],C![7,-23,2],C![11,-5975,15],C![11,-1206,15]]; // minimal model
 
magma: [C![-2,-1,1],C![-2,1,1],C![-1,-3,1],C![-1,3,1],C![0,-3,1],C![0,3,1],C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,1,0],C![4,-53,3],C![4,53,3],C![7,-333,2],C![7,333,2],C![11,-4769,15],C![11,4769,15]]; // 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
\((-2 : 4 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2\) \(0.560794\) \(\infty\)
\((1 : -2 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.349279\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.236667\) \(\infty\)
Generator $D_0$ Height Order
\((-2 : 4 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2\) \(0.560794\) \(\infty\)
\((1 : -2 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.349279\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.236667\) \(\infty\)
Generator $D_0$ Height Order
\((-2 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3xz^2 + z^3\) \(0.560794\) \(\infty\)
\((1 : -1 : 1) - (1 : 1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + xz^2 - z^3\) \(0.349279\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 + z^3\) \(0.236667\) \(\infty\)

2-torsion field: 6.2.5077568.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.039916 \)
Real period: \( 18.56975 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.741242 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(79337\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 495 T + 79337 T^{2} )\)

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);