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Properties

Label 363.a.11979.1
Conductor 363
Discriminant -11979
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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The Jacobian of this curve is isogenous to that of the quotient of the modular curve $X_0(33)$ by the involution $W_3$ (see [MR:1373390]), which has discriminant $3\cdot 11^9$.

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 4, 2, 0, 1], R![1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 4, 2, 0, 1]), R([1, 0, 1]))
 

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 363 \)  =  \( 3 \cdot 11^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-11979\)  =  \( -1 \cdot 3^{2} \cdot 11^{3} \)

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);
 

\( I_2 \)  =  \(1376\)  =  \( 2^{5} \cdot 43 \)
\( I_4 \)  =  \(-49088\)  =  \( -1 \cdot 2^{6} \cdot 13 \cdot 59 \)
\( I_6 \)  =  \(-33691712\)  =  \( -1 \cdot 2^{6} \cdot 19 \cdot 103 \cdot 269 \)
\( I_{10} \)  =  \(-49065984\)  =  \( -1 \cdot 2^{12} \cdot 3^{2} \cdot 11^{3} \)
\( J_2 \)  =  \(172\)  =  \( 2^{2} \cdot 43 \)
\( J_4 \)  =  \(1744\)  =  \( 2^{4} \cdot 109 \)
\( J_6 \)  =  \(45841\)  =  \( 45841 \)
\( J_8 \)  =  \(1210779\)  =  \( 3^{2} \cdot 29 \cdot 4639 \)
\( J_{10} \)  =  \(-11979\)  =  \( -1 \cdot 3^{2} \cdot 11^{3} \)
\( g_1 \)  =  \(-150536645632/11979\)
\( g_2 \)  =  \(-8874253312/11979\)
\( g_3 \)  =  \(-1356160144/11979\)
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,-34,4],C![-1,-1,1],C![0,-1,1],C![0,0,1],C![1,0,0]];
 

All rational points: (-1 : -34 : 4), (-1 : -1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : 0 : 0)

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

Number of rational Weierstrass points: \(3\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(2\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 1.0

Real period: 18.970595995348840324264371503

Tamagawa numbers: 2 (p = 3), 2 (p = 11)

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

Torsion: \(\Z/{2}\Z \times \Z/{10}\Z\)

2-torsion field: 3.1.44.1

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 11.a3
  Elliptic curve 33.a2

Endomorphisms

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)an order of index \(3\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).