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Properties

Label 249.a.6723.1
Conductor $249$
Discriminant $-6723$
Mordell-Weil group \(\Z/{28}\Z\)
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

$y^2 + (x^3 + 1)y = -x^5 + x^3 + x^2 + 3x + 2$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -x^5z + x^3z^3 + x^2z^4 + 3xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 4x^5 + 6x^3 + 4x^2 + 12x + 9$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 3, 1, 1, 0, -1]), R([1, 0, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 3, 1, 1, 0, -1], R![1, 0, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([9, 12, 4, 6, 0, -4, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(249\) \(=\) \( 3 \cdot 83 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-6723\) \(=\) \( - 3^{4} \cdot 83 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1932\) \(=\)  \( 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
\( I_4 \)  \(=\) \(87897\) \(=\)  \( 3 \cdot 83 \cdot 353 \)
\( I_6 \)  \(=\) \(65765571\) \(=\)  \( 3 \cdot 17 \cdot 89 \cdot 14489 \)
\( I_{10} \)  \(=\) \(860544\) \(=\)  \( 2^{7} \cdot 3^{4} \cdot 83 \)
\( J_2 \)  \(=\) \(483\) \(=\)  \( 3 \cdot 7 \cdot 23 \)
\( J_4 \)  \(=\) \(6058\) \(=\)  \( 2 \cdot 13 \cdot 233 \)
\( J_6 \)  \(=\) \(-161212\) \(=\)  \( - 2^{2} \cdot 41 \cdot 983 \)
\( J_8 \)  \(=\) \(-28641190\) \(=\)  \( - 2 \cdot 5 \cdot 97 \cdot 29527 \)
\( J_{10} \)  \(=\) \(6723\) \(=\)  \( 3^{4} \cdot 83 \)
\( g_1 \)  \(=\) \(324526850403/83\)
\( g_2 \)  \(=\) \(25281736298/249\)
\( g_3 \)  \(=\) \(-4178776252/747\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

Copy content sage:C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
Copy content magma:IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$
Copy content magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (0 : 1 : 1),\, (0 : -2 : 1),\, (3 : -14 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (0 : 1 : 1),\, (0 : -2 : 1),\, (3 : -14 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : 0 : 1),\, (0 : -3 : 1),\, (0 : 3 : 1),\, (3 : 0 : 1)\)

Copy content magma:[C![-1,0,1],C![0,-2,1],C![0,1,1],C![1,-1,0],C![1,0,0],C![3,-14,1]]; // minimal model
 
Copy content magma:[C![-1,0,1],C![0,-3,1],C![0,3,1],C![1,-1,0],C![1,1,0],C![3,0,1]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

Copy content magma:#Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

Copy content 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: \(\Z/{28}\Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 - 5xz - 6z^2\) \(=\) \(0,\) \(4y\) \(=\) \(-15xz^2 - 14z^3\) \(0\) \(28\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 - 5xz - 6z^2\) \(=\) \(0,\) \(4y\) \(=\) \(-15xz^2 - 14z^3\) \(0\) \(28\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(2x^2 - 5xz - 6z^2\) \(=\) \(0,\) \(4y\) \(=\) \(x^3 - 30xz^2 - 27z^3\) \(0\) \(28\)

2-torsion field: 4.2.1328.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 25.78370 \)
Tamagawa product: \( 4 \)
Torsion order:\( 28 \)
Leading coefficient: \( 0.131549 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number L-factor Cluster picture Tame reduction?
\(3\) \(1\) \(4\) \(4\) \(-1\) \(( 1 - T )( 1 + 3 T + 3 T^{2} )\) yes
\(83\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 83 T^{2} )\) yes

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?
\(2\) 2.30.3 yes
\(7\) not computed yes

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Copy content 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 \(\Q\).

Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

Copy content magma:HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

Copy content magma:HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);