Properties

Label 12595.a.138545.1
Conductor $12595$
Discriminant $138545$
Mordell-Weil group \(\Z/{6}\Z\)
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

$y^2 + (x^3 + 1)y = x^6 - 6x^4 + x^3 + 4x^2 - 2x$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^6 - 6x^4z^2 + x^3z^3 + 4x^2z^4 - 2xz^5$ (dehomogenize, simplify)
$y^2 = 5x^6 - 24x^4 + 6x^3 + 16x^2 - 8x + 1$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); Cmin = HyperellipticCurve(R([0, -2, 4, 1, -6, 0, 1]), R([1, 0, 0, 1])) # minimal equation
 
Copy content magma:R<x> := PolynomialRing(Rationals()); Cmin := HyperellipticCurve(R![0, -2, 4, 1, -6, 0, 1], R![1, 0, 0, 1]); // minimal equation
 
Copy content sage:Csim = HyperellipticCurve(R([1, -8, 16, 6, -24, 0, 5])); Csim # simplified equation
 
Copy content magma:Csim, pi := SimplifiedModel(Cmin); Csim; // simplified equation
 

Invariants

Conductor: \( N \)  \(=\)  \(12595\) \(=\) \( 5 \cdot 11 \cdot 229 \)
Copy content magma:Conductor(LSeries(Cmin)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(138545\) \(=\) \( 5 \cdot 11^{2} \cdot 229 \)
Copy content magma:Discriminant(Cmin); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2580\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 43 \)
\( I_4 \)  \(=\) \(173697\) \(=\)  \( 3 \cdot 57899 \)
\( I_6 \)  \(=\) \(127604433\) \(=\)  \( 3 \cdot 11 \cdot 59 \cdot 65539 \)
\( I_{10} \)  \(=\) \(17733760\) \(=\)  \( 2^{7} \cdot 5 \cdot 11^{2} \cdot 229 \)
\( J_2 \)  \(=\) \(645\) \(=\)  \( 3 \cdot 5 \cdot 43 \)
\( J_4 \)  \(=\) \(10097\) \(=\)  \( 23 \cdot 439 \)
\( J_6 \)  \(=\) \(145561\) \(=\)  \( 13 \cdot 11197 \)
\( J_8 \)  \(=\) \(-2015641\) \(=\)  \( - 1361 \cdot 1481 \)
\( J_{10} \)  \(=\) \(138545\) \(=\)  \( 5 \cdot 11^{2} \cdot 229 \)
\( g_1 \)  \(=\) \(22326907280625/27709\)
\( g_2 \)  \(=\) \(541877970825/27709\)
\( g_3 \)  \(=\) \(12111403005/27709\)

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

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

Automorphism group

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

Rational points

All points: \((0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1)\)
All points: \((0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1)\)
All points: \((-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1)\)

Copy content magma:[Cmin![-1,0,1],Cmin![0,-1,1],Cmin![0,0,1]]; // minimal model
 
Copy content magma:[Csim![-1,0,1],Csim![0,-1,1],Csim![0,1,1]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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

This curve is locally solvable everywhere.

Copy content magma:f,h:=HyperellipticPolynomials(Cmin); 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: \(\Z/{6}\Z\)

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

Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 0 : 1) - D_\infty\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(6\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 0 : 1) - D_\infty\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(6\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 1 : 1) - D_\infty\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + z^3\) \(0\) \(6\)

2-torsion field: 6.6.6555125.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 19.98460 \)
Tamagawa product: \( 2 \)
Torsion order:\( 6 \)
Leading coefficient: \( 1.110255 \)
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?
\(5\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 3 T + 5 T^{2} )\) yes
\(11\) \(1\) \(2\) \(2\) \(1\) \(( 1 + T )( 1 - 3 T + 11 T^{2} )\) yes
\(229\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 - 14 T + 229 T^{2} )\) 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?
\(2\) 2.60.1 yes
\(3\) 3.80.1 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(Cmin);
 

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(Cmin); HeuristicEndomorphismDescription(Cmin); HeuristicEndomorphismFieldOfDefinition(Cmin);
 

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