Properties

Label 2916.a.139968.1
Conductor 2916
Discriminant -139968
Sato-Tate group $J(E_1)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, -6, 5, -3, 1], R![1, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, -6, 5, -3, 1]), R([1, 1, 1]))
 

$y^2 + (x^2 + x + 1)y = x^6 - 3x^5 + 5x^4 - 6x^3 + x$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 2916 \)  =  \( 2^{2} \cdot 3^{6} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-139968\)  =  \( -1 \cdot 2^{6} \cdot 3^{7} \)

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);
 

\( I_2 \)  =  \(-1944\)  =  \( -1 \cdot 2^{3} \cdot 3^{5} \)
\( I_4 \)  =  \(453924\)  =  \( 2^{2} \cdot 3^{5} \cdot 467 \)
\( I_6 \)  =  \(-384120792\)  =  \( -1 \cdot 2^{3} \cdot 3^{5} \cdot 11^{2} \cdot 23 \cdot 71 \)
\( I_{10} \)  =  \(-573308928\)  =  \( -1 \cdot 2^{18} \cdot 3^{7} \)
\( J_2 \)  =  \(-243\)  =  \( -1 \cdot 3^{5} \)
\( J_4 \)  =  \(-2268\)  =  \( -1 \cdot 2^{2} \cdot 3^{4} \cdot 7 \)
\( J_6 \)  =  \(314496\)  =  \( 2^{7} \cdot 3^{3} \cdot 7 \cdot 13 \)
\( J_8 \)  =  \(-20391588\)  =  \( -1 \cdot 2^{2} \cdot 3^{9} \cdot 7 \cdot 37 \)
\( J_{10} \)  =  \(-139968\)  =  \( -1 \cdot 2^{6} \cdot 3^{7} \)
\( g_1 \)  =  \(387420489/64\)
\( g_2 \)  =  \(-3720087/16\)
\( g_3 \)  =  \(-132678\)
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![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,1,0]];
 

All rational points: (0 : -1 : 1), (0 : 0 : 1), (1 : -2 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 1 : 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: \(0\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 1.0

Real period: 19.520681040807033229434261445

Tamagawa numbers: 9 (p = 2), 3 (p = 3)

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

Torsion: \(\Z/{3}\Z \times \Z/{9}\Z\)

2-torsion field: 6.0.139968.1

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $J(E_1)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 54.b2
  Elliptic curve 54.a2

Endomorphisms

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

Endomorphism ring over \(\Q\):
\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\)\(\simeq\)an Eichler order of index \(9\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)