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Properties

Label 1148.a.47068.1
Conductor $1148$
Discriminant $-47068$
Mordell-Weil group \(\Z/{10}\Z\)
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

$y^2 + (x^2 + x + 1)y = x^5 + 2x^4 - 5x^3 + x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2 + z^3)y = x^5z + 2x^4z^2 - 5x^3z^3 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 9x^4 - 18x^3 + 3x^2 + 6x + 1$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, -5, 2, 1]), R([1, 1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, -5, 2, 1], R![1, 1, 1]);
 
Copy content sage:X = HyperellipticCurve(R([1, 6, 3, -18, 9, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1148\) \(=\) \( 2^{2} \cdot 7 \cdot 41 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-47068\) \(=\) \( - 2^{2} \cdot 7 \cdot 41^{2} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1236\) \(=\)  \( 2^{2} \cdot 3 \cdot 103 \)
\( I_4 \)  \(=\) \(129537\) \(=\)  \( 3^{2} \cdot 37 \cdot 389 \)
\( I_6 \)  \(=\) \(36025137\) \(=\)  \( 3^{2} \cdot 1907 \cdot 2099 \)
\( I_{10} \)  \(=\) \(-6024704\) \(=\)  \( - 2^{9} \cdot 7 \cdot 41^{2} \)
\( J_2 \)  \(=\) \(309\) \(=\)  \( 3 \cdot 103 \)
\( J_4 \)  \(=\) \(-1419\) \(=\)  \( - 3 \cdot 11 \cdot 43 \)
\( J_6 \)  \(=\) \(31221\) \(=\)  \( 3^{2} \cdot 3469 \)
\( J_8 \)  \(=\) \(1908432\) \(=\)  \( 2^{4} \cdot 3^{2} \cdot 29 \cdot 457 \)
\( J_{10} \)  \(=\) \(-47068\) \(=\)  \( - 2^{2} \cdot 7 \cdot 41^{2} \)
\( g_1 \)  \(=\) \(-2817036000549/47068\)
\( g_2 \)  \(=\) \(41865649551/47068\)
\( g_3 \)  \(=\) \(-2981012301/47068\)

  (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

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -26 : 4)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -26 : 4)\)
All points: \((1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : 0 : 4)\)

Copy content magma:[C![-1,-26,4],C![0,-1,1],C![0,0,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![-1,0,4],C![0,-1,1],C![0,1,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

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: \(\Z/{10}\Z\)

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

Generator $D_0$ Height Order
\((-1 : -26 : 4) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (4x + z)\) \(=\) \(0,\) \(8y\) \(=\) \(13xz^2\) \(0\) \(10\)
Generator $D_0$ Height Order
\((-1 : -26 : 4) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (4x + z)\) \(=\) \(0,\) \(8y\) \(=\) \(13xz^2\) \(0\) \(10\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x (4x + z)\) \(=\) \(0,\) \(8y\) \(=\) \(x^2z + 27xz^2 + z^3\) \(0\) \(10\)

2-torsion field: \(\Q(\sqrt{10 +2 \sqrt{-7}})\)

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 22.53131 \)
Tamagawa product: \( 2 \)
Torsion order:\( 10 \)
Leading coefficient: \( 0.450626 \)
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?
\(2\) \(2\) \(2\) \(1\) \(1^*\) \(1 + T^{2}\) yes
\(7\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 2 T + 7 T^{2} )\) yes
\(41\) \(1\) \(2\) \(2\) \(-1\) \(( 1 - T )( 1 + 8 T + 41 T^{2} )\) yes

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?
\(2\) 2.90.3 yes
\(5\) not computed yes

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);