# 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

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); $$\Delta$$ = $$81920$$ = $$2^{14} \cdot 5$$

### 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: 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 Tamagawa numbers: 6 (p = 2), 1 (p = 5) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{12}\Z$$

### 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 $$\Z \times \Z [\sqrt{-1}]$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q(\sqrt{-1})$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \C$$