↑

Properties

Label 12228.a.97824.1
Conductor $12228$
Discriminant $97824$
Mordell-Weil group \(\Z \oplus \Z/{4}\Z\)
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

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(12228\) \(=\) \( 2^{2} \cdot 3 \cdot 1019 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(97824\) \(=\) \( 2^{5} \cdot 3 \cdot 1019 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(628\) \(=\)  \( 2^{2} \cdot 157 \)
\( I_4 \)  \(=\) \(12409\) \(=\)  \( 12409 \)
\( I_6 \)  \(=\) \(2409901\) \(=\)  \( 13 \cdot 197 \cdot 941 \)
\( I_{10} \)  \(=\) \(12521472\) \(=\)  \( 2^{12} \cdot 3 \cdot 1019 \)
\( J_2 \)  \(=\) \(157\) \(=\)  \( 157 \)
\( J_4 \)  \(=\) \(510\) \(=\)  \( 2 \cdot 3 \cdot 5 \cdot 17 \)
\( J_6 \)  \(=\) \(-1964\) \(=\)  \( - 2^{2} \cdot 491 \)
\( J_8 \)  \(=\) \(-142112\) \(=\)  \( - 2^{5} \cdot 4441 \)
\( J_{10} \)  \(=\) \(97824\) \(=\)  \( 2^{5} \cdot 3 \cdot 1019 \)
\( g_1 \)  \(=\) \(95388992557/97824\)
\( g_2 \)  \(=\) \(328940905/16304\)
\( g_3 \)  \(=\) \(-12102659/24456\)

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

Copy content magma:[C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,1,0]]; // 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: \(\Z \oplus \Z/{4}\Z\)

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

Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.550206\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(4\)
Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.550206\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(4\)
Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + xz^2 - 4z^3\) \(0.550206\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + xz^2\) \(0\) \(4\)

2-torsion field: 6.0.1794287808.3

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.550206 \)
Real period: \( 7.359269 \)
Tamagawa product: \( 4 \)
Torsion order:\( 4 \)
Leading coefficient: \( 1.012278 \)
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?
\(2\) \(2\) \(5\) \(4\) \(-1^*\) \(( 1 - T )( 1 + T )\) yes
\(3\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\) yes
\(1019\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 - 4 T + 1019 T^{2} )\) 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?
\(2\) 2.15.1 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(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);