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Properties

Label 62208.l.373248.1
Conductor $62208$
Discriminant $373248$
Mordell-Weil group \(\Z \oplus \Z\)
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

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

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

Invariants

Conductor: \( N \)  \(=\)  \(62208\) \(=\) \( 2^{8} \cdot 3^{5} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(373248\) \(=\) \( 2^{9} \cdot 3^{6} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(36\) \(=\)  \( 2^{2} \cdot 3^{2} \)
\( I_4 \)  \(=\) \(54\) \(=\)  \( 2 \cdot 3^{3} \)
\( I_6 \)  \(=\) \(828\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 23 \)
\( I_{10} \)  \(=\) \(-192\) \(=\)  \( - 2^{6} \cdot 3 \)
\( J_2 \)  \(=\) \(108\) \(=\)  \( 2^{2} \cdot 3^{3} \)
\( J_4 \)  \(=\) \(162\) \(=\)  \( 2 \cdot 3^{4} \)
\( J_6 \)  \(=\) \(-7236\) \(=\)  \( - 2^{2} \cdot 3^{3} \cdot 67 \)
\( J_8 \)  \(=\) \(-201933\) \(=\)  \( - 3^{6} \cdot 277 \)
\( J_{10} \)  \(=\) \(-373248\) \(=\)  \( - 2^{9} \cdot 3^{6} \)
\( g_1 \)  \(=\) \(-39366\)
\( g_2 \)  \(=\) \(-2187/4\)
\( g_3 \)  \(=\) \(1809/8\)

  (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

Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((-2 : 1 : 3)\) \((-2 : -10 : 3)\) \((-1 : -23 : 4)\) \((-1 : -25 : 4)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((-2 : 1 : 3)\) \((-2 : -10 : 3)\) \((-1 : -23 : 4)\) \((-1 : -25 : 4)\)
Known points
\((1 : -2 : 0)\) \((1 : 2 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -2 : 1)\) \((-1 : 2 : 1)\)
\((-1 : -2 : 4)\) \((-1 : 2 : 4)\) \((-2 : -11 : 3)\) \((-2 : 11 : 3)\)

Copy content magma:[C![-2,-10,3],C![-2,1,3],C![-1,-25,4],C![-1,-23,4],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,1,0]]; // minimal model
 
Copy content magma:[C![-2,-11,3],C![-2,11,3],C![-1,-2,4],C![-1,2,4],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-2,0],C![1,2,0]]; // 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: \(\Z \oplus \Z\)

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

Generator $D_0$ Height Order
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.420134\) \(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3\) \(0.041264\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.420134\) \(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3\) \(0.041264\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : -2 : 1) + (0 : -1 : 1) - (1 : -2 : 0) - (1 : 2 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0.420134\) \(\infty\)
\((0 : 1 : 1) - (1 : -2 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(2x^3 + xz^2 + z^3\) \(0.041264\) \(\infty\)

2-torsion field: 6.2.1492992.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.016985 \)
Real period: \( 13.39983 \)
Tamagawa product: \( 6 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.365653 \)
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?
\(2\) \(8\) \(9\) \(2\) \(1^*\) \(1 + T + 2 T^{2}\) no
\(3\) \(5\) \(6\) \(3\) \(1^*\) \(1\) no

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
\(3\) 3.540.6 no

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);