Properties

Label 226133.a.226133.1
Conductor $226133$
Discriminant $226133$
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

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 4, -6, 2, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 4, -6, 2, 1], R![1]);
 
sage: X = HyperellipticCurve(R([1, -4, 16, -24, 8, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(226133\) \(=\) \( 226133 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(226133\) \(=\) \( 226133 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(192\) \(=\)  \( 2^{6} \cdot 3 \)
\( I_4 \)  \(=\) \(10368\) \(=\)  \( 2^{7} \cdot 3^{4} \)
\( I_6 \)  \(=\) \(376272\) \(=\)  \( 2^{4} \cdot 3^{3} \cdot 13 \cdot 67 \)
\( I_{10} \)  \(=\) \(904532\) \(=\)  \( 2^{2} \cdot 226133 \)
\( J_2 \)  \(=\) \(96\) \(=\)  \( 2^{5} \cdot 3 \)
\( J_4 \)  \(=\) \(-1344\) \(=\)  \( - 2^{6} \cdot 3 \cdot 7 \)
\( J_6 \)  \(=\) \(6320\) \(=\)  \( 2^{4} \cdot 5 \cdot 79 \)
\( J_8 \)  \(=\) \(-299904\) \(=\)  \( - 2^{7} \cdot 3 \cdot 11 \cdot 71 \)
\( J_{10} \)  \(=\) \(226133\) \(=\)  \( 226133 \)
\( g_1 \)  \(=\) \(8153726976/226133\)
\( g_2 \)  \(=\) \(-1189085184/226133\)
\( g_3 \)  \(=\) \(58245120/226133\)

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
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\) \((-1 : 3 : 1)\)
\((-1 : -4 : 1)\) \((2 : 5 : 1)\) \((2 : -6 : 1)\) \((1 : -6 : 4)\) \((1 : -58 : 4)\) \((4 : -69 : 9)\)
\((7 : 140 : 1)\) \((7 : -141 : 1)\) \((4 : -660 : 9)\)
Known points
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -1 : 1)\) \((-1 : 3 : 1)\)
\((-1 : -4 : 1)\) \((2 : 5 : 1)\) \((2 : -6 : 1)\) \((1 : -6 : 4)\) \((1 : -58 : 4)\) \((4 : -69 : 9)\)
\((7 : 140 : 1)\) \((7 : -141 : 1)\) \((4 : -660 : 9)\)
Known points
\((1 : 0 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : -1 : 1)\) \((1 : 1 : 1)\) \((-1 : -7 : 1)\)
\((-1 : 7 : 1)\) \((2 : -11 : 1)\) \((2 : 11 : 1)\) \((1 : -52 : 4)\) \((1 : 52 : 4)\) \((7 : -281 : 1)\)
\((7 : 281 : 1)\) \((4 : -591 : 9)\) \((4 : 591 : 9)\)

magma: [C![-1,-4,1],C![-1,3,1],C![0,-1,1],C![0,0,1],C![1,-58,4],C![1,-6,4],C![1,-1,1],C![1,0,0],C![1,0,1],C![2,-6,1],C![2,5,1],C![4,-660,9],C![4,-69,9],C![7,-141,1],C![7,140,1]]; // minimal model
 
magma: [C![-1,-7,1],C![-1,7,1],C![0,-1,1],C![0,1,1],C![1,-52,4],C![1,52,4],C![1,-1,1],C![1,0,0],C![1,1,1],C![2,-11,1],C![2,11,1],C![4,-591,9],C![4,591,9],C![7,-281,1],C![7,281,1]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 + z^3\) \(0.961045\) \(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.282858\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.251453\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 + z^3\) \(0.961045\) \(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.282858\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.251453\) \(\infty\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-6xz^2 + 3z^3\) \(0.961045\) \(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.282858\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.251453\) \(\infty\)

2-torsion field: 5.1.3618128.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.067360 \)
Real period: \( 18.46537 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.243832 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(226133\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 330 T + 226133 T^{2} )\)

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?
\(2\) 2.6.1 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

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);