Properties

Label 1083.b.390963.1
Conductor 1083
Discriminant -390963
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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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-6, -33, -14, 95, -50, 3, -1], R![1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, -33, -14, 95, -50, 3, -1]), R([1]))

$y^2 + y = -x^6 + 3x^5 - 50x^4 + 95x^3 - 14x^2 - 33x - 6$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 1083 \)  =  \( 3 \cdot 19^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-390963\)  =  \( -1 \cdot 3 \cdot 19^{4} \)

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);

\( I_2 \)  =  \(601760\)  =  \( 2^{5} \cdot 5 \cdot 3761 \)
\( I_4 \)  =  \(31128254272\)  =  \( 2^{6} \cdot 486378973 \)
\( I_6 \)  =  \(4413239365215232\)  =  \( 2^{10} \cdot 17 \cdot 97 \cdot 2857 \cdot 914801 \)
\( I_{10} \)  =  \(-1601384448\)  =  \( -1 \cdot 2^{12} \cdot 3 \cdot 19^{4} \)
\( J_2 \)  =  \(75220\)  =  \( 2^{2} \cdot 5 \cdot 3761 \)
\( J_4 \)  =  \(-88500632\)  =  \( -1 \cdot 2^{3} \cdot 11 \cdot 19 \cdot 41 \cdot 1291 \)
\( J_6 \)  =  \(98386538568\)  =  \( 2^{3} \cdot 3 \cdot 19^{3} \cdot 597673 \)
\( J_8 \)  =  \(-107931608328616\)  =  \( -1 \cdot 2^{3} \cdot 19^{2} \cdot 37 \cdot 1010065961 \)
\( J_{10} \)  =  \(-390963\)  =  \( -1 \cdot 3 \cdot 19^{4} \)
\( g_1 \)  =  \(-2408056349828975363200000/390963\)
\( g_2 \)  =  \(1982406707133537344000/20577\)
\( g_3 \)  =  \(-27053302090985600/19\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
\(\mathrm{Aut}(X)\)\(\simeq\) \(V_4 \) (GAP id : [4,2])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(V_4 \) (GAP id : [4,2])

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 except over $\R$.

magma: [];

There are no rational points.

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank: \(0\)

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

2-Selmer rank: \(1\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: twice a square

Tamagawa numbers: 1 (p = 3), 1 (p = 19)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.0.69312.1

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 19.a1
  Elliptic curve 57.b1

Endomorphisms

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

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)an order of index \(2\) 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 \(\Q\).