Properties

Label 56473.a.56473.1
Conductor 56473
Discriminant 56473
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![1, -2, 0, 1, -2], R![0, 1, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -2, 0, 1, -2]), R([0, 1, 1, 1]))
 

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 56473 \)  =  \( 56473 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(56473\)  =  \( 56473 \)

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);
 

\( I_2 \)  =  \(-1304\)  =  \( -1 \cdot 2^{3} \cdot 163 \)
\( I_4 \)  =  \(27364\)  =  \( 2^{2} \cdot 6841 \)
\( I_6 \)  =  \(-15809432\)  =  \( -1 \cdot 2^{3} \cdot 173 \cdot 11423 \)
\( I_{10} \)  =  \(231313408\)  =  \( 2^{12} \cdot 56473 \)
\( J_2 \)  =  \(-163\)  =  \( -1 \cdot 163 \)
\( J_4 \)  =  \(822\)  =  \( 2 \cdot 3 \cdot 137 \)
\( J_6 \)  =  \(4516\)  =  \( 2^{2} \cdot 1129 \)
\( J_8 \)  =  \(-352948\)  =  \( -1 \cdot 2^{2} \cdot 88237 \)
\( J_{10} \)  =  \(56473\)  =  \( 56473 \)
\( g_1 \)  =  \(-115063617043/56473\)
\( g_2 \)  =  \(-3559874034/56473\)
\( g_3 \)  =  \(119985604/56473\)
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![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-7,2],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![1,0,2],C![3,-35,1],C![3,-4,1],C![5,-279,6],C![5,-224,4],C![5,-176,6],C![5,-81,4]];
 

Known rational points: (-1 : 0 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 1 : 1), (1 : -7 : 2), (1 : -2 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (1 : 0 : 2), (3 : -35 : 1), (3 : -4 : 1), (5 : -279 : 6), (5 : -224 : 4), (5 : -176 : 6), (5 : -81 : 4)

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: 0.0378755005364

Real period: 18.143500677999799895673634113

Tamagawa numbers: 1 (p = 56473)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.2.3614272.1

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 \(\Q\).