Properties

Label 1104.b.141312.1
Conductor 1104
Discriminant -141312
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

Related objects

Learn more about

Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-46, 0, 29, 0, -3, 0, -1], R![0, 1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-46, 0, 29, 0, -3, 0, -1]), R([0, 1, 0, 1]))

$y^2 + (x^3 + x)y = -x^6 - 3x^4 + 29x^2 - 46$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
\( N \)  =  \( 1104 \)  =  \( 2^{4} \cdot 3 \cdot 23 \)
magma: Discriminant(C); Factorization(Integers()!$1);
\( \Delta \)  =  \(-141312\)  =  \( -1 \cdot 2^{11} \cdot 3 \cdot 23 \)

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);

\( I_2 \)  =  \(-113760\)  =  \( -1 \cdot 2^{5} \cdot 3^{2} \cdot 5 \cdot 79 \)
\( I_4 \)  =  \(602799168\)  =  \( 2^{6} \cdot 3 \cdot 19 \cdot 149 \cdot 1109 \)
\( I_6 \)  =  \(-27791527951872\)  =  \( -1 \cdot 2^{9} \cdot 3^{2} \cdot 503 \cdot 11990353 \)
\( I_{10} \)  =  \(-578813952\)  =  \( -1 \cdot 2^{23} \cdot 3 \cdot 23 \)
\( J_2 \)  =  \(-14220\)  =  \( -1 \cdot 2^{2} \cdot 3^{2} \cdot 5 \cdot 79 \)
\( J_4 \)  =  \(2146192\)  =  \( 2^{4} \cdot 31 \cdot 4327 \)
\( J_6 \)  =  \(16790479872\)  =  \( 2^{10} \cdot 3 \cdot 23 \cdot 71 \cdot 3347 \)
\( J_8 \)  =  \(-60841690970176\)  =  \( -1 \cdot 2^{6} \cdot 950651421409 \)
\( J_{10} \)  =  \(-141312\)  =  \( -1 \cdot 2^{11} \cdot 3 \cdot 23 \)
\( g_1 \)  =  \(189267815942240625/46\)
\( g_2 \)  =  \(2008843709918625/46\)
\( g_3 \)  =  \(-24026098775400\)
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 $\Q_{2}$.

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: \(2\)

magma: HasSquareSha(Jacobian(C));

Order of Ш*: twice a square

Tamagawa numbers: 1 (p = 2), 1 (p = 3), 1 (p = 23)

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

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

2-torsion field: 8.4.2808152064.3

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 46.a1
  Elliptic curve 24.a1

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 \(\Q\).