↑

Properties

Label 200704.d.401408.1
Conductor $200704$
Discriminant $401408$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z/{2}\Z\)
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

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

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

Invariants

Conductor: \( N \)  \(=\)  \(200704\) \(=\) \( 2^{12} \cdot 7^{2} \)
Copy content magma:Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(200704,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(401408\) \(=\) \( 2^{13} \cdot 7^{2} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(451\) \(=\)  \( 11 \cdot 41 \)
\( I_4 \)  \(=\) \(1330\) \(=\)  \( 2 \cdot 5 \cdot 7 \cdot 19 \)
\( I_6 \)  \(=\) \(187628\) \(=\)  \( 2^{2} \cdot 7 \cdot 6701 \)
\( I_{10} \)  \(=\) \(49\) \(=\)  \( 7^{2} \)
\( J_2 \)  \(=\) \(1804\) \(=\)  \( 2^{2} \cdot 11 \cdot 41 \)
\( J_4 \)  \(=\) \(121414\) \(=\)  \( 2 \cdot 17 \cdot 3571 \)
\( J_6 \)  \(=\) \(10025348\) \(=\)  \( 2^{2} \cdot 2506337 \)
\( J_8 \)  \(=\) \(836092099\) \(=\)  \( 836092099 \)
\( J_{10} \)  \(=\) \(401408\) \(=\)  \( 2^{13} \cdot 7^{2} \)
\( g_1 \)  \(=\) \(18658757027251/392\)
\( g_2 \)  \(=\) \(5568886892657/3136\)
\( g_3 \)  \(=\) \(509791452137/6272\)

  (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

Known points: \((1 : 0 : 0),\, (1 : -1 : 1),\, (1 : 1 : 1),\, (1 : -2 : 2),\, (1 : 2 : 2)\)
Known points: \((1 : 0 : 0),\, (1 : -1 : 1),\, (1 : 1 : 1),\, (1 : -2 : 2),\, (1 : 2 : 2)\)
Known points: \((1 : 0 : 0),\, (1 : -1/2 : 1),\, (1 : 1/2 : 1),\, (1 : -1 : 2),\, (1 : 1 : 2)\)

Copy content magma:[C![1,-2,2],C![1,-1,1],C![1,0,0],C![1,1,1],C![1,2,2]]; // minimal model
 
Copy content magma:[C![1,-1,2],C![1,-1/2,1],C![1,0,0],C![1,1/2,1],C![1,1,2]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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: \(\Z \oplus \Z \oplus \Z/{2}\Z\)

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

Generator $D_0$ Height Order
\((1 : -2 : 2) + (1 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (2x - z)\) \(=\) \(0,\) \(2y\) \(=\) \(5xz^2 - 3z^3\) \(0.772055\) \(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.501386\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(2x^2 - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -2 : 2) + (1 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (2x - z)\) \(=\) \(0,\) \(2y\) \(=\) \(5xz^2 - 3z^3\) \(0.772055\) \(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.501386\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(2x^2 - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -1 : 2) + (1 : 1/2 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (2x - z)\) \(=\) \(0,\) \(2y\) \(=\) \(5/2xz^2 - 3/2z^3\) \(0.772055\) \(\infty\)
\((1 : -1/2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-1/2z^3\) \(0.501386\) \(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(2x^2 - z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: 6.6.1229312.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(3\)
Regulator: \( 0.336432 \)
Real period: \( 16.21416 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.727483 \)
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?
\(2\) \(12\) \(13\) \(2\) \(-1^*\) \(1\) no
\(7\) \(2\) \(2\) \(1\) \(-1\) \(1 + 7 T^{2}\) 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?
\(2\) 2.120.5 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);