Properties

Label 42439.a.42439.1
Conductor 42439
Discriminant -42439
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![0, 1, -1, -1, 3, -3, 1], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -1, -1, 3, -3, 1]), R([1]))
 

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 42439 \)  =  \( 31 \cdot 37^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-42439\)  =  \( -1 \cdot 31 \cdot 37^{2} \)

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);
 

\( I_2 \)  =  \(-2016\)  =  \( -1 \cdot 2^{5} \cdot 3^{2} \cdot 7 \)
\( I_4 \)  =  \(50496\)  =  \( 2^{6} \cdot 3 \cdot 263 \)
\( I_6 \)  =  \(-42974208\)  =  \( -1 \cdot 2^{10} \cdot 3^{2} \cdot 4663 \)
\( I_{10} \)  =  \(-173830144\)  =  \( -1 \cdot 2^{12} \cdot 31 \cdot 37^{2} \)
\( J_2 \)  =  \(-252\)  =  \( -1 \cdot 2^{2} \cdot 3^{2} \cdot 7 \)
\( J_4 \)  =  \(2120\)  =  \( 2^{3} \cdot 5 \cdot 53 \)
\( J_6 \)  =  \(744\)  =  \( 2^{3} \cdot 3 \cdot 31 \)
\( J_8 \)  =  \(-1170472\)  =  \( -1 \cdot 2^{3} \cdot 146309 \)
\( J_{10} \)  =  \(-42439\)  =  \( -1 \cdot 31 \cdot 37^{2} \)
\( g_1 \)  =  \(1016255020032/42439\)
\( g_2 \)  =  \(33926376960/42439\)
\( g_3 \)  =  \(-1524096/1369\)
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![-2,-15,1],C![-2,14,1],C![-1,-3,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,-1,1],C![1,0,1],C![1,1,0],C![2,-3,1],C![2,2,1],C![3,-15,1],C![3,14,1]];
 

Known rational points: (-2 : -15 : 1), (-2 : 14 : 1), (-1 : -3 : 1), (-1 : 2 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 1), (1 : 1 : 0), (2 : -3 : 1), (2 : 2 : 1), (3 : -15 : 1), (3 : 14 : 1)

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank*: \(3\)

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

2-Selmer rank: \(3\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 0.0298105686716

Real period: 19.960990222519150448056079746

Tamagawa numbers: 1 (p = 31), 1 (p = 37)

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

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.4.2716096.1

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 37.a1
  Elliptic curve 1147.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 \(\Q\).