Properties

Label 101291.a.101291.1
Conductor $101291$
Discriminant $-101291$
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

$y^2 + (x^3 + x^2 + x + 1)y = 4x^3 + 5x^2 - 3x$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = 4x^3z^3 + 5x^2z^4 - 3xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 3x^4 + 20x^3 + 23x^2 - 10x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 5, 4]), R([1, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 5, 4], R![1, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([1, -10, 23, 20, 3, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(101291\) \(=\) \( 199 \cdot 509 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-101291\) \(=\) \( - 199 \cdot 509 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(64\) \(=\)  \( 2^{6} \)
\( I_4 \)  \(=\) \(67360\) \(=\)  \( 2^{5} \cdot 5 \cdot 421 \)
\( I_6 \)  \(=\) \(-4243856\) \(=\)  \( - 2^{4} \cdot 265241 \)
\( I_{10} \)  \(=\) \(-405164\) \(=\)  \( - 2^{2} \cdot 199 \cdot 509 \)
\( J_2 \)  \(=\) \(32\) \(=\)  \( 2^{5} \)
\( J_4 \)  \(=\) \(-11184\) \(=\)  \( - 2^{4} \cdot 3 \cdot 233 \)
\( J_6 \)  \(=\) \(571408\) \(=\)  \( 2^{4} \cdot 71 \cdot 503 \)
\( J_8 \)  \(=\) \(-26699200\) \(=\)  \( - 2^{6} \cdot 5^{2} \cdot 11 \cdot 37 \cdot 41 \)
\( J_{10} \)  \(=\) \(-101291\) \(=\)  \( - 199 \cdot 509 \)
\( g_1 \)  \(=\) \(-33554432/101291\)
\( g_2 \)  \(=\) \(366477312/101291\)
\( g_3 \)  \(=\) \(-585121792/101291\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -2 : 1)\) \((-1 : 2 : 1)\)
\((1 : 1 : 2)\) \((-2 : 2 : 1)\) \((-2 : 3 : 1)\) \((1 : -16 : 2)\) \((3 : 16 : 2)\) \((-2 : 41 : 3)\)
\((-2 : -54 : 3)\) \((3 : -81 : 2)\) \((-21 : 2576 : 10)\) \((-21 : 3375 : 10)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -2 : 1)\) \((-1 : 2 : 1)\)
\((1 : 1 : 2)\) \((-2 : 2 : 1)\) \((-2 : 3 : 1)\) \((1 : -16 : 2)\) \((3 : 16 : 2)\) \((-2 : 41 : 3)\)
\((-2 : -54 : 3)\) \((3 : -81 : 2)\) \((-21 : 2576 : 10)\) \((-21 : 3375 : 10)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-2 : -1 : 1)\) \((-2 : 1 : 1)\)
\((-1 : -4 : 1)\) \((-1 : 4 : 1)\) \((1 : -17 : 2)\) \((1 : 17 : 2)\) \((-2 : -95 : 3)\) \((-2 : 95 : 3)\)
\((3 : -97 : 2)\) \((3 : 97 : 2)\) \((-21 : -799 : 10)\) \((-21 : 799 : 10)\)

magma: [C![-21,2576,10],C![-21,3375,10],C![-2,-54,3],C![-2,2,1],C![-2,3,1],C![-2,41,3],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-16,2],C![1,-1,0],C![1,0,0],C![1,1,2],C![3,-81,2],C![3,16,2]]; // minimal model
 
magma: [C![-21,-799,10],C![-21,799,10],C![-2,-95,3],C![-2,-1,1],C![-2,1,1],C![-2,95,3],C![-1,-4,1],C![-1,4,1],C![0,-1,1],C![0,1,1],C![1,-17,2],C![1,-1,0],C![1,1,0],C![1,17,2],C![3,-97,2],C![3,97,2]]; // 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
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.494311\) \(\infty\)
\((-2 : 2 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.260271\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.341158\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.494311\) \(\infty\)
\((-2 : 2 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.260271\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.341158\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + xz^2 - z^3\) \(0.494311\) \(\infty\)
\((-2 : -1 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z - xz^2 + z^3\) \(0.260271\) \(\infty\)
\((0 : -1 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + x^2z + xz^2 - z^3\) \(0.341158\) \(\infty\)

2-torsion field: 6.0.1620656.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.039401 \)
Real period: \( 20.81300 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.820069 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(199\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 5 T + 199 T^{2} )\)
\(509\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 30 T + 509 T^{2} )\)

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);