Properties

Label 65814.a.394884.1
Conductor 65814
Discriminant 394884
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

Learn more about

Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, -3, -1, 0, 1, 1], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, -3, -1, 0, 1, 1]), R([1, 0, 0, 1]))
 

$y^2 + (x^3 + 1)y = x^5 + x^4 - x^2 - 3x + 2$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 65814 \)  =  \( 2 \cdot 3 \cdot 7 \cdot 1567 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(394884\)  =  \( 2^{2} \cdot 3^{2} \cdot 7 \cdot 1567 \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  =  \(-3800\)  =  \( -1 \cdot 2^{3} \cdot 5^{2} \cdot 19 \)
\( I_4 \)  =  \(220132\)  =  \( 2^{2} \cdot 11 \cdot 5003 \)
\( I_6 \)  =  \(-317768344\)  =  \( -1 \cdot 2^{3} \cdot 39721043 \)
\( I_{10} \)  =  \(1617444864\)  =  \( 2^{14} \cdot 3^{2} \cdot 7 \cdot 1567 \)
\( J_2 \)  =  \(-475\)  =  \( -1 \cdot 5^{2} \cdot 19 \)
\( J_4 \)  =  \(7108\)  =  \( 2^{2} \cdot 1777 \)
\( J_6 \)  =  \(1044\)  =  \( 2^{2} \cdot 3^{2} \cdot 29 \)
\( J_8 \)  =  \(-12754891\)  =  \( -1 \cdot 607 \cdot 21013 \)
\( J_{10} \)  =  \(394884\)  =  \( 2^{2} \cdot 3^{2} \cdot 7 \cdot 1567 \)
\( g_1 \)  =  \(-24180654296875/394884\)
\( g_2 \)  =  \(-190444421875/98721\)
\( g_3 \)  =  \(6543125/10969\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) \(C_2 \) (GAP id : [2,1])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(C_2 \) (GAP id : [2,1])

Rational points

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);
 

This curve is locally solvable everywhere.

magma: [C![-51,38270,14],C![-51,91637,14],C![-4,-33,3],C![-4,70,3],C![-3,10,1],C![-3,16,1],C![-2,3,1],C![-2,4,1],C![-1,-2,1],C![-1,2,1],C![0,-2,1],C![0,1,1],C![1,-11,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![1,2,2],C![3,-46,2],C![3,11,2],C![4,-114,3],C![4,23,3]];
 

Known rational points: (-51 : 38270 : 14), (-51 : 91637 : 14), (-4 : -33 : 3), (-4 : 70 : 3), (-3 : 10 : 1), (-3 : 16 : 1), (-2 : 3 : 1), (-2 : 4 : 1), (-1 : -2 : 1), (-1 : 2 : 1), (0 : -2 : 1), (0 : 1 : 1), (1 : -11 : 2), (1 : -2 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 0 : 1), (1 : 2 : 2), (3 : -46 : 2), (3 : 11 : 2), (4 : -114 : 3), (4 : 23 : 3)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank*: \(3\)

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);
 

2-Selmer rank: \(3\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 0.0119224382662

Real period: 16.312532861135774343644688883

Tamagawa numbers: 2 (p = 2), 2 (p = 3), 1 (p = 7), 1 (p = 1567)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
 

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.2.702016.1

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition

Simple over \(\overline{\Q}\)

Endomorphisms

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 \(\Q\).