Properties

Label 640.a.81920.2
Conductor 640
Discriminant 81920
Sato-Tate group $N(G_{1,3})$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{CM} \times \Q\)
\(\overline{\Q}\)-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![-20, 0, 13, 0, -3], R![0, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-20, 0, 13, 0, -3]), R([0, 0, 0, 1]))
 

$y^2 + x^3y = -3x^4 + 13x^2 - 20$

Invariants

magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(640,2),R![1]>*])); Factorization($1);
 
\( N \)  =  \( 640 \)  =  \( 2^{7} \cdot 5 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(81920\)  =  \( 2^{14} \cdot 5 \)

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);
 

\( I_2 \)  =  \(29184\)  =  \( 2^{9} \cdot 3 \cdot 19 \)
\( I_4 \)  =  \(150528\)  =  \( 2^{10} \cdot 3 \cdot 7^{2} \)
\( I_6 \)  =  \(1460207616\)  =  \( 2^{16} \cdot 3 \cdot 7 \cdot 1061 \)
\( I_{10} \)  =  \(335544320\)  =  \( 2^{26} \cdot 5 \)
\( J_2 \)  =  \(3648\)  =  \( 2^{6} \cdot 3 \cdot 19 \)
\( J_4 \)  =  \(552928\)  =  \( 2^{5} \cdot 37 \cdot 467 \)
\( J_6 \)  =  \(111431680\)  =  \( 2^{12} \cdot 5 \cdot 5441 \)
\( J_8 \)  =  \(25193348864\)  =  \( 2^{8} \cdot 73 \cdot 379 \cdot 3557 \)
\( J_{10} \)  =  \(81920\)  =  \( 2^{14} \cdot 5 \)
\( g_1 \)  =  \(39432490647552/5\)
\( g_2 \)  =  \(1638374321664/5\)
\( g_3 \)  =  \(18102076416\)
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![-2,4,1],C![1,-1,0],C![1,0,0],C![2,-4,1]];
 

All rational points: (-2 : 4 : 1), (1 : -1 : 0), (1 : 0 : 0), (2 : -4 : 1)

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

Number of rational Weierstrass points: \(2\)

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: 7.4056741771287726282328124212

Tamagawa numbers: 6 (p = 2), 1 (p = 5)

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

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

2-torsion field: 4.0.320.1

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(G_{1,3})$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\times\mathrm{SU}(2)\)

Decomposition

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 32.a4
  Elliptic curve 20.a4

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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)

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

Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(4\) in \(\Z \times \Z [\sqrt{-1}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-1}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)