Properties

Label 5911.b.5911.1
Conductor 5911
Discriminant -5911
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![0, 0, -1, 1, -2], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -1, 1, -2]), R([1, 1, 0, 1]))
 

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 5911 \)  =  \( 23 \cdot 257 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-5911\)  =  \( -1 \cdot 23 \cdot 257 \)

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);
 

\( I_2 \)  =  \(-312\)  =  \( -1 \cdot 2^{3} \cdot 3 \cdot 13 \)
\( I_4 \)  =  \(-12060\)  =  \( -1 \cdot 2^{2} \cdot 3^{2} \cdot 5 \cdot 67 \)
\( I_6 \)  =  \(79848\)  =  \( 2^{3} \cdot 3^{2} \cdot 1109 \)
\( I_{10} \)  =  \(-24211456\)  =  \( -1 \cdot 2^{12} \cdot 23 \cdot 257 \)
\( J_2 \)  =  \(-39\)  =  \( -1 \cdot 3 \cdot 13 \)
\( J_4 \)  =  \(189\)  =  \( 3^{3} \cdot 7 \)
\( J_6 \)  =  \(1085\)  =  \( 5 \cdot 7 \cdot 31 \)
\( J_8 \)  =  \(-19509\)  =  \( -1 \cdot 3 \cdot 7 \cdot 929 \)
\( J_{10} \)  =  \(-5911\)  =  \( -1 \cdot 23 \cdot 257 \)
\( g_1 \)  =  \(90224199/5911\)
\( g_2 \)  =  \(11211291/5911\)
\( g_3 \)  =  \(-1650285/5911\)
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![-3,11,1],C![-3,18,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-44,3],C![2,-9,3],C![2,-7,1],C![2,-4,1]];
 

Known rational points: (-3 : 11 : 1), (-3 : 18 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -2 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (2 : -44 : 3), (2 : -9 : 3), (2 : -7 : 1), (2 : -4 : 1)

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank*: \(2\)

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

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 0.0136452684536

Real period: 19.752007613340955791056414200

Tamagawa numbers: 1 (p = 23), 1 (p = 257)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.0.206529873031.1

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 \(\Q\).