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Properties

Label 1116.a.214272.1
Conductor $1116$
Discriminant $-214272$
Mordell-Weil group \(\Z/{39}\Z\)
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

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

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 1, 2, 1]), R([1, 0, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 1, 2, 1], R![1, 0, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([1, -4, 4, 10, 4, 0, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1116\) \(=\) \( 2^{2} \cdot 3^{2} \cdot 31 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-214272\) \(=\) \( - 2^{8} \cdot 3^{3} \cdot 31 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(52\) \(=\)  \( 2^{2} \cdot 13 \)
\( I_4 \)  \(=\) \(22201\) \(=\)  \( 149^{2} \)
\( I_6 \)  \(=\) \(238285\) \(=\)  \( 5 \cdot 47657 \)
\( I_{10} \)  \(=\) \(-27426816\) \(=\)  \( - 2^{15} \cdot 3^{3} \cdot 31 \)
\( J_2 \)  \(=\) \(13\) \(=\)  \( 13 \)
\( J_4 \)  \(=\) \(-918\) \(=\)  \( - 2 \cdot 3^{3} \cdot 17 \)
\( J_6 \)  \(=\) \(36\) \(=\)  \( 2^{2} \cdot 3^{2} \)
\( J_8 \)  \(=\) \(-210564\) \(=\)  \( - 2^{2} \cdot 3^{2} \cdot 5849 \)
\( J_{10} \)  \(=\) \(-214272\) \(=\)  \( - 2^{8} \cdot 3^{3} \cdot 31 \)
\( g_1 \)  \(=\) \(-371293/214272\)
\( g_2 \)  \(=\) \(37349/3968\)
\( g_3 \)  \(=\) \(-169/5952\)

  (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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((1 : -3 : 1)\)
All points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((1 : -3 : 1)\)
All points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -2 : 1)\) \((-1 : 2 : 1)\)
\((1 : -4 : 1)\) \((1 : 4 : 1)\)

Copy content magma:[C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,1,1]]; // minimal model
 
Copy content magma:[C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-4,1],C![1,-1,0],C![1,1,0],C![1,4,1]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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: \(\Z/{39}\Z\)

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

Generator $D_0$ Height Order
\((-1 : 1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0\) \(39\)
Generator $D_0$ Height Order
\((-1 : 1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0\) \(39\)
Generator $D_0$ Height Order
\((-1 : 2 : 1) + (1 : -4 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 4xz^2 - z^3\) \(0\) \(39\)

2-torsion field: 6.0.53568.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 16.98409 \)
Tamagawa product: \( 39 \)
Torsion order:\( 39 \)
Leading coefficient: \( 0.435489 \)
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?
\(2\) \(2\) \(8\) \(13\) \(-1^*\) \(( 1 - T )( 1 + T )\) yes
\(3\) \(2\) \(3\) \(3\) \(1\) \(1 + T + T^{2}\) yes
\(31\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 7 T + 31 T^{2} )\) 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?
\(2\) 2.10.1 no
\(3\) 3.80.1 yes
\(13\) 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);
 

Additional information

This example of a curve of genus 2 whose Jacobian has a rational 39-torsion point was discovered by Noam Elkies (see this page).