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Properties

Label 2309.a.2309.1
Conductor $2309$
Discriminant $2309$
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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 0, -2, 1]), R([1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 0, -2, 1], R![1, 1, 1]);
 
sage: X = HyperellipticCurve(R([1, -2, 3, 2, -7, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(2309\) \(=\) \( 2309 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(2309\) \(=\) \( 2309 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(20\) \(=\)  \( 2^{2} \cdot 5 \)
\( I_4 \)  \(=\) \(1417\) \(=\)  \( 13 \cdot 109 \)
\( I_6 \)  \(=\) \(-17979\) \(=\)  \( - 3 \cdot 13 \cdot 461 \)
\( I_{10} \)  \(=\) \(295552\) \(=\)  \( 2^{7} \cdot 2309 \)
\( J_2 \)  \(=\) \(5\) \(=\)  \( 5 \)
\( J_4 \)  \(=\) \(-58\) \(=\)  \( - 2 \cdot 29 \)
\( J_6 \)  \(=\) \(332\) \(=\)  \( 2^{2} \cdot 83 \)
\( J_8 \)  \(=\) \(-426\) \(=\)  \( - 2 \cdot 3 \cdot 71 \)
\( J_{10} \)  \(=\) \(2309\) \(=\)  \( 2309 \)
\( g_1 \)  \(=\) \(3125/2309\)
\( g_2 \)  \(=\) \(-7250/2309\)
\( g_3 \)  \(=\) \(8300/2309\)

  (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

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (1 : -2 : 1),\, (3 : -46 : 4),\, (3 : -102 : 4)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (1 : -2 : 1),\, (3 : -46 : 4),\, (3 : -102 : 4)\)
All points: \((1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : -1 : 1),\, (1 : 1 : 1),\, (3 : -56 : 4),\, (3 : 56 : 4)\)

magma: [C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,1],C![1,0,0],C![3,-102,4],C![3,-46,4]]; // minimal model
 
magma: [C![0,-1,1],C![0,1,1],C![1,-1,1],C![1,1,1],C![1,0,0],C![3,-56,4],C![3,56,4]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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
\((0 : 0 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.013966\) \(\infty\)
Generator $D_0$ Height Order
\((0 : 0 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.013966\) \(\infty\)
Generator $D_0$ Height Order
\((0 : 1 : 1) + (1 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - xz^2 + z^3\) \(0.013966\) \(\infty\)

2-torsion field: 5.1.36944.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.013966 \)
Real period: \( 21.86168 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.305334 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2309\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 59 T + 2309 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.6.1 no

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);