Properties

Label 20532.a.82128.1
Conductor $20532$
Discriminant $82128$
Mordell-Weil group \(\Z/{3}\Z\)
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

$y^2 + (x^2 + x)y = x^6 - x^5 + 4x^4 + x^3 + 2x^2 + 7x + 3$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^6 - x^5z + 4x^4z^2 + x^3z^3 + 2x^2z^4 + 7xz^5 + 3z^6$ (dehomogenize, simplify)
$y^2 = 4x^6 - 4x^5 + 17x^4 + 6x^3 + 9x^2 + 28x + 12$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 7, 2, 1, 4, -1, 1]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 7, 2, 1, 4, -1, 1], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([12, 28, 9, 6, 17, -4, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(20532\) \(=\) \( 2^{2} \cdot 3 \cdot 29 \cdot 59 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(82128\) \(=\) \( 2^{4} \cdot 3 \cdot 29 \cdot 59 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(9116\) \(=\)  \( 2^{2} \cdot 43 \cdot 53 \)
\( I_4 \)  \(=\) \(5271409\) \(=\)  \( 11 \cdot 13 \cdot 191 \cdot 193 \)
\( I_6 \)  \(=\) \(11983616247\) \(=\)  \( 3 \cdot 151 \cdot 227 \cdot 116537 \)
\( I_{10} \)  \(=\) \(-10512384\) \(=\)  \( - 2^{11} \cdot 3 \cdot 29 \cdot 59 \)
\( J_2 \)  \(=\) \(2279\) \(=\)  \( 43 \cdot 53 \)
\( J_4 \)  \(=\) \(-3232\) \(=\)  \( - 2^{5} \cdot 101 \)
\( J_6 \)  \(=\) \(6416\) \(=\)  \( 2^{4} \cdot 401 \)
\( J_8 \)  \(=\) \(1044060\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 17401 \)
\( J_{10} \)  \(=\) \(-82128\) \(=\)  \( - 2^{4} \cdot 3 \cdot 29 \cdot 59 \)
\( g_1 \)  \(=\) \(-61478268295547399/82128\)
\( g_2 \)  \(=\) \(2391026255078/5133\)
\( g_3 \)  \(=\) \(-2082730241/5133\)

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

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

Number of rational Weierstrass points: \(0\)

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: \(\Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 6.2.8655532008768.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 6.461730 \)
Tamagawa product: \( 1 \)
Torsion order:\( 3 \)
Leading coefficient: \( 0.717970 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(4\) \(1\) \(( 1 + T )^{2}\)
\(3\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(29\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 29 T^{2} )\)
\(59\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 9 T + 59 T^{2} )\)

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?
\(2\) 2.20.2 no
\(3\) 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 \(\overline{\Q}\)

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);