↑

Properties

Label 456247.a.456247.1
Conductor $456247$
Discriminant $-456247$
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

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = x^5 - 8x^4 + 3x^3 + 7x^2 + 2x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = x^5z - 8x^4z^2 + 3x^3z^3 + 7x^2z^4 + 2xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 30x^4 + 14x^3 + 29x^2 + 10x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 7, 3, -8, 1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 7, 3, -8, 1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 10, 29, 14, -30, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(456247\) \(=\) \( 11 \cdot 19 \cdot 37 \cdot 59 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-456247\) \(=\) \( - 11 \cdot 19 \cdot 37 \cdot 59 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(8228\) \(=\)  \( 2^{2} \cdot 11^{2} \cdot 17 \)
\( I_4 \)  \(=\) \(33241\) \(=\)  \( 13 \cdot 2557 \)
\( I_6 \)  \(=\) \(94691661\) \(=\)  \( 3 \cdot 281 \cdot 112327 \)
\( I_{10} \)  \(=\) \(-58399616\) \(=\)  \( - 2^{7} \cdot 11 \cdot 19 \cdot 37 \cdot 59 \)
\( J_2 \)  \(=\) \(2057\) \(=\)  \( 11^{2} \cdot 17 \)
\( J_4 \)  \(=\) \(174917\) \(=\)  \( 174917 \)
\( J_6 \)  \(=\) \(19623641\) \(=\)  \( 73 \cdot 268817 \)
\( J_8 \)  \(=\) \(2442468162\) \(=\)  \( 2 \cdot 3 \cdot 1481 \cdot 274867 \)
\( J_{10} \)  \(=\) \(-456247\) \(=\)  \( - 11 \cdot 19 \cdot 37 \cdot 59 \)
\( g_1 \)  \(=\) \(-3347948534700187/41477\)
\( g_2 \)  \(=\) \(-138401950309271/41477\)
\( g_3 \)  \(=\) \(-7548410123419/41477\)

  (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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-2 : -440 : 11),\, (-2 : -641 : 11)\)
Known points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-2 : -440 : 11),\, (-2 : -641 : 11)\)
Known points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-2 : -201 : 11),\, (-2 : 201 : 11)\)

magma: [C![-2,-641,11],C![-2,-440,11],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![-2,-201,11],C![-2,201,11],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,1,0]]; // 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: \(\Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2\) \(0.675867\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.541760\) \(\infty\)
Generator $D_0$ Height Order
\(2 \cdot(0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(2xz^2\) \(0.675867\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.541760\) \(\infty\)
Generator $D_0$ Height Order
\(2 \cdot(0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 5xz^2 + z^3\) \(0.675867\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 - z^3\) \(0.541760\) \(\infty\)

2-torsion field: 6.4.29199808.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.320469 \)
Real period: \( 6.344468 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 2.033209 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(11\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 6 T + 11 T^{2} )\)
\(19\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 19 T^{2} )\)
\(37\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 6 T + 37 T^{2} )\)
\(59\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 59 T^{2} )\)

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);