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Properties

Label 132206.b.264412.1
Conductor $132206$
Discriminant $-264412$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z\)
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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -2, 0, 4, 2]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -2, 0, 4, 2], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, -4, -8, 2, 16, 8, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(132206\) \(=\) \( 2 \cdot 66103 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-264412\) \(=\) \( - 2^{2} \cdot 66103 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(276\) \(=\)  \( 2^{2} \cdot 3 \cdot 23 \)
\( I_4 \)  \(=\) \(32505\) \(=\)  \( 3 \cdot 5 \cdot 11 \cdot 197 \)
\( I_6 \)  \(=\) \(-45267\) \(=\)  \( - 3 \cdot 79 \cdot 191 \)
\( I_{10} \)  \(=\) \(-33844736\) \(=\)  \( - 2^{9} \cdot 66103 \)
\( J_2 \)  \(=\) \(69\) \(=\)  \( 3 \cdot 23 \)
\( J_4 \)  \(=\) \(-1156\) \(=\)  \( - 2^{2} \cdot 17^{2} \)
\( J_6 \)  \(=\) \(27348\) \(=\)  \( 2^{2} \cdot 3 \cdot 43 \cdot 53 \)
\( J_8 \)  \(=\) \(137669\) \(=\)  \( 7 \cdot 71 \cdot 277 \)
\( J_{10} \)  \(=\) \(-264412\) \(=\)  \( - 2^{2} \cdot 66103 \)
\( g_1 \)  \(=\) \(-1564031349/264412\)
\( g_2 \)  \(=\) \(94939101/66103\)
\( g_3 \)  \(=\) \(-32550957/66103\)

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

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

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((-2 : 1 : 1)\) \((1 : -3 : 1)\) \((-2 : 6 : 1)\) \((-1 : 8 : 4)\) \((-5 : 55 : 1)\)
\((-5 : 69 : 1)\) \((1 : -69 : 6)\) \((-1 : -71 : 4)\) \((-6 : 74 : 1)\) \((-6 : 141 : 1)\) \((1 : -148 : 6)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((-2 : 1 : 1)\) \((1 : -3 : 1)\) \((-2 : 6 : 1)\) \((-1 : 8 : 4)\) \((-5 : 55 : 1)\)
\((-5 : 69 : 1)\) \((1 : -69 : 6)\) \((-1 : -71 : 4)\) \((-6 : 74 : 1)\) \((-6 : 141 : 1)\) \((1 : -148 : 6)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -2 : 1)\) \((-1 : 2 : 1)\)
\((1 : -4 : 1)\) \((1 : 4 : 1)\) \((-2 : -5 : 1)\) \((-2 : 5 : 1)\) \((-5 : -14 : 1)\) \((-5 : 14 : 1)\)
\((-6 : -67 : 1)\) \((-6 : 67 : 1)\) \((-1 : -79 : 4)\) \((-1 : 79 : 4)\) \((1 : -79 : 6)\) \((1 : 79 : 6)\)

magma: [C![-6,74,1],C![-6,141,1],C![-5,55,1],C![-5,69,1],C![-2,1,1],C![-2,6,1],C![-1,-71,4],C![-1,-1,1],C![-1,1,1],C![-1,8,4],C![0,-1,1],C![0,0,1],C![1,-148,6],C![1,-69,6],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,1,1]]; // minimal model
 
magma: [C![-6,-67,1],C![-6,67,1],C![-5,-14,1],C![-5,14,1],C![-2,-5,1],C![-2,5,1],C![-1,-79,4],C![-1,-2,1],C![-1,2,1],C![-1,79,4],C![0,-1,1],C![0,1,1],C![1,-79,6],C![1,79,6],C![1,-4,1],C![1,-1,0],C![1,1,0],C![1,4,1]]; // 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: \(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(2x^2z\) \(0.554061\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.266044\) \(\infty\)
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 2z^3\) \(0.226795\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(2x^2z\) \(0.554061\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.266044\) \(\infty\)
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 2z^3\) \(0.226795\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 4x^2z + z^3\) \(0.554061\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0.266044\) \(\infty\)
\((-1 : -2 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 3z^3\) \(0.226795\) \(\infty\)

2-torsion field: 6.4.4230592.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.027620 \)
Real period: \( 17.77731 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.982052 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(66103\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 397 T + 66103 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .

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);