# Properties

 Label 810.a.196830.1 Conductor 810 Discriminant -196830 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![-8, 94, -297, 20, 15, 1], R![1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-8, 94, -297, 20, 15, 1]), R([1, 1]))

$y^2 + (x + 1)y = x^5 + 15x^4 + 20x^3 - 297x^2 + 94x - 8$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$810$$ = $$2 \cdot 3^{4} \cdot 5$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-196830$$ = $$-1 \cdot 2 \cdot 3^{9} \cdot 5$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$1238400$$ = $$2^{7} \cdot 3^{2} \cdot 5^{2} \cdot 43$$ $$I_4$$ = $$13269432960$$ = $$2^{7} \cdot 3^{6} \cdot 5 \cdot 7 \cdot 17 \cdot 239$$ $$I_6$$ = $$4967784069073344$$ = $$2^{6} \cdot 3^{5} \cdot 17 \cdot 18790032941$$ $$I_{10}$$ = $$-806215680$$ = $$-1 \cdot 2^{13} \cdot 3^{9} \cdot 5$$ $$J_2$$ = $$154800$$ = $$2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 43$$ $$J_4$$ = $$860236740$$ = $$2^{2} \cdot 3^{3} \cdot 5 \cdot 11 \cdot 97 \cdot 1493$$ $$J_6$$ = $$5905731060081$$ = $$3^{4} \cdot 31 \cdot 2351943871$$ $$J_8$$ = $$43549979813677800$$ = $$2^{3} \cdot 3^{10} \cdot 5^{2} \cdot 7753 \cdot 475637$$ $$J_{10}$$ = $$-196830$$ = $$-1 \cdot 2 \cdot 3^{9} \cdot 5$$ $$g_1$$ = $$-451609936896000000000$$ $$g_2$$ = $$-16212110811776000000$$ $$g_3$$ = $$-2156977131869584000/3$$
Alternative geometric invariants: G2

## Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$C_2$$ (GAP id : [2,1]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$C_2$$ (GAP id : [2,1])

## 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![1,0,0]];

All rational points: (1 : 0 : 0)

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

Number of rational Weierstrass points: $$1$$

## 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: 1 (p = 2), 4 (p = 3), 1 (p = 5) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{2}\Z$$

### 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 30.a1
Elliptic curve 27.a1

### Endomorphisms

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ an order of index $$3$$ 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 order of index $$27$$ in $$\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q(\sqrt{-3})$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \C$$