↑

Properties

Label 741.a.28899.1
Conductor $741$
Discriminant $-28899$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{8}\Z\)
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

$y^2 + (x + 1)y = -3x^5 - x^4 + 2x^2 + x$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = -3x^5z - x^4z^2 + 2x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = -12x^5 - 4x^4 + 9x^2 + 6x + 1$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 2, 0, -1, -3]), R([1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 2, 0, -1, -3], R![1, 1]);
 
Copy content sage:X = HyperellipticCurve(R([1, 6, 9, 0, -4, -12]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(741\) \(=\) \( 3 \cdot 13 \cdot 19 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-28899\) \(=\) \( - 3^{2} \cdot 13^{2} \cdot 19 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(576\) \(=\)  \( 2^{6} \cdot 3^{2} \)
\( I_4 \)  \(=\) \(-840\) \(=\)  \( - 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
\( I_6 \)  \(=\) \(740385\) \(=\)  \( 3^{2} \cdot 5 \cdot 16453 \)
\( I_{10} \)  \(=\) \(115596\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 13^{2} \cdot 19 \)
\( J_2 \)  \(=\) \(288\) \(=\)  \( 2^{5} \cdot 3^{2} \)
\( J_4 \)  \(=\) \(3596\) \(=\)  \( 2^{2} \cdot 29 \cdot 31 \)
\( J_6 \)  \(=\) \(-38169\) \(=\)  \( - 3^{2} \cdot 4241 \)
\( J_8 \)  \(=\) \(-5980972\) \(=\)  \( - 2^{2} \cdot 19 \cdot 78697 \)
\( J_{10} \)  \(=\) \(28899\) \(=\)  \( 3^{2} \cdot 13^{2} \cdot 19 \)
\( g_1 \)  \(=\) \(220150628352/3211\)
\( g_2 \)  \(=\) \(9544531968/3211\)
\( g_3 \)  \(=\) \(-351765504/3211\)

  (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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (-1 : -9 : 3)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -1 : 1),\, (-1 : -9 : 3)\)
All points: \((1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 0 : 1),\, (-1 : 0 : 3)\)

Copy content magma:[C![-1,-9,3],C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![-1,0,3],C![0,-1,1],C![0,1,1],C![1,0,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(3\)

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: \(\Z/{2}\Z \oplus \Z/{8}\Z\)

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

Generator $D_0$ Height Order
\((-1 : -9 : 3) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (3x + z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
\((0 : -1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(8\)
Generator $D_0$ Height Order
\((-1 : -9 : 3) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (3x + z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
\((0 : -1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(8\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \((x - z) (3x + z)\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)
\((0 : -1 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0\) \(8\)

2-torsion field: 3.1.76.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 18.75684 \)
Tamagawa product: \( 4 \)
Torsion order:\( 16 \)
Leading coefficient: \( 0.293075 \)
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?
\(3\) \(1\) \(2\) \(2\) \(1\) \(( 1 + T )( 1 + 3 T^{2} )\) yes
\(13\) \(1\) \(2\) \(2\) \(1\) \(( 1 + T )( 1 + 2 T + 13 T^{2} )\) yes
\(19\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 19 T^{2} )\) 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?
\(2\) 2.120.3 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);