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Properties

Label 10086.b.413526.1
Conductor $10086$
Discriminant $413526$
Mordell-Weil group \(\Z/{6}\Z\)
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

$y^2 + (x^3 + x)y = -x^6 - 4x^5 - 7x^4 - 6x^3 + 3x + 3$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^6 - 4x^5z - 7x^4z^2 - 6x^3z^3 + 3xz^5 + 3z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 - 16x^5 - 26x^4 - 24x^3 + x^2 + 12x + 12$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 3, 0, -6, -7, -4, -1]), R([0, 1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 3, 0, -6, -7, -4, -1], R![0, 1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([12, 12, 1, -24, -26, -16, -3]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10086\) \(=\) \( 2 \cdot 3 \cdot 41^{2} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(413526\) \(=\) \( 2 \cdot 3 \cdot 41^{3} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1208\) \(=\)  \( 2^{3} \cdot 151 \)
\( I_4 \)  \(=\) \(136120\) \(=\)  \( 2^{3} \cdot 5 \cdot 41 \cdot 83 \)
\( I_6 \)  \(=\) \(71419827\) \(=\)  \( 3 \cdot 41 \cdot 101 \cdot 5749 \)
\( I_{10} \)  \(=\) \(1654104\) \(=\)  \( 2^{3} \cdot 3 \cdot 41^{3} \)
\( J_2 \)  \(=\) \(604\) \(=\)  \( 2^{2} \cdot 151 \)
\( J_4 \)  \(=\) \(-7486\) \(=\)  \( - 2 \cdot 19 \cdot 197 \)
\( J_6 \)  \(=\) \(-3619151\) \(=\)  \( -3619151 \)
\( J_8 \)  \(=\) \(-560501850\) \(=\)  \( - 2 \cdot 3 \cdot 5^{2} \cdot 29 \cdot 269 \cdot 479 \)
\( J_{10} \)  \(=\) \(413526\) \(=\)  \( 2 \cdot 3 \cdot 41^{3} \)
\( g_1 \)  \(=\) \(40193395584512/206763\)
\( g_2 \)  \(=\) \(-824765797952/206763\)
\( g_3 \)  \(=\) \(-660162095608/206763\)

  (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

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

Copy content magma:[]; // minimal model
 
Copy content magma:[]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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: \(\Z/{6}\Z\)

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

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - 2z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - 2z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 3xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3xz^2 - 4z^3\) \(0\) \(6\)

2-torsion field: 6.0.30984192.4

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(3\)
Regulator: \( 1 \)
Real period: \( 5.766529 \)
Tamagawa product: \( 2 \)
Torsion order:\( 6 \)
Leading coefficient: \( 1.281450 \)
Analytic order of Ш: \( 4 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(1\) \(1\) \(1\) \(-1^*\) \(( 1 - T )( 1 + 2 T^{2} )\) yes
\(3\) \(1\) \(1\) \(1\) \(-1\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\) yes
\(41\) \(2\) \(3\) \(2\) \(1\) \(1 - 6 T + 41 T^{2}\) 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?
\(2\) 2.15.1 yes
\(3\) 3.80.1 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);