Properties

Label 784.a.43904.1
Conductor 784
Discriminant -43904
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

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Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![56, 0, 27, 0, 4], R![0, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([56, 0, 27, 0, 4]), R([0, 1, 0, 1]))
 

$y^2 + (x^3 + x)y = 4x^4 + 27x^2 + 56$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 784 \)  =  \( 2^{4} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-43904\)  =  \( -1 \cdot 2^{7} \cdot 7^{3} \)

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);
 

\( I_2 \)  =  \(-85152\)  =  \( -1 \cdot 2^{5} \cdot 3 \cdot 887 \)
\( I_4 \)  =  \(48000\)  =  \( 2^{7} \cdot 3 \cdot 5^{3} \)
\( I_6 \)  =  \(-1337035008\)  =  \( -1 \cdot 2^{8} \cdot 3 \cdot 1740931 \)
\( I_{10} \)  =  \(-179830784\)  =  \( -1 \cdot 2^{19} \cdot 7^{3} \)
\( J_2 \)  =  \(-10644\)  =  \( -1 \cdot 2^{2} \cdot 3 \cdot 887 \)
\( J_4 \)  =  \(4720114\)  =  \( 2 \cdot 7 \cdot 233 \cdot 1447 \)
\( J_6 \)  =  \(-2790613504\)  =  \( -1 \cdot 2^{9} \cdot 7^{2} \cdot 41 \cdot 2713 \)
\( J_8 \)  =  \(1855953490895\)  =  \( 5 \cdot 7^{2} \cdot 727 \cdot 10419973 \)
\( J_{10} \)  =  \(-43904\)  =  \( -1 \cdot 2^{7} \cdot 7^{3} \)
\( g_1 \)  =  \(1067368445729034408/343\)
\( g_2 \)  =  \(6352710665144931/49\)
\( g_3 \)  =  \(50408453477952/7\)
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 everywhere.

magma: [C![1,-1,0],C![1,0,0]];
 

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

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*: square

Regulator: 1.0

Real period: 6.9311170700356541844063992761

Tamagawa numbers: 2 (p = 2), 3 (p = 7)

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

Torsion: \(\Z/{12}\Z\)

2-torsion field: 4.0.392.1

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 56.a4
  Elliptic curve 14.a6

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 \(\Q\).