# Properties

 Label 980.a.878080.1 Conductor 980 Discriminant -878080 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

Show commands for: Magma / SageMath

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 1, -4, 2, -4, 1, -1], R![1, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 1, -4, 2, -4, 1, -1]), R([1, 0, 0, 1]))

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

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$980$$ = $$2^{2} \cdot 5 \cdot 7^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-878080$$ = $$-1 \cdot 2^{9} \cdot 5 \cdot 7^{3}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-5016$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 11 \cdot 19$$ $$I_4$$ = $$202980$$ = $$2^{2} \cdot 3 \cdot 5 \cdot 17 \cdot 199$$ $$I_6$$ = $$-333605784$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 2129 \cdot 6529$$ $$I_{10}$$ = $$-3596615680$$ = $$-1 \cdot 2^{21} \cdot 5 \cdot 7^{3}$$ $$J_2$$ = $$-627$$ = $$-1 \cdot 3 \cdot 11 \cdot 19$$ $$J_4$$ = $$14266$$ = $$2 \cdot 7 \cdot 1019$$ $$J_6$$ = $$-359660$$ = $$-1 \cdot 2^{2} \cdot 5 \cdot 7^{2} \cdot 367$$ $$J_8$$ = $$5497016$$ = $$2^{3} \cdot 7^{2} \cdot 37 \cdot 379$$ $$J_{10}$$ = $$-878080$$ = $$-1 \cdot 2^{9} \cdot 5 \cdot 7^{3}$$ $$g_1$$ = $$96903107471907/878080$$ $$g_2$$ = $$251175228777/62720$$ $$g_3$$ = $$144278343/896$$
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 except over $\R$ and $\Q_{2}$.

magma: [];

There are no rational points.

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: $$1$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: N/A (p = 2), 1 (p = 5), 3 (p = 7) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

### 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 14.a6
Elliptic curve 70.a4

### Endomorphisms

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

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

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.