# Properties

 Label 726.a.1452.1 Conductor 726 Discriminant -1452 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![0, -1, -2, 6, 2, 2], R![1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -2, 6, 2, 2]), R([1, 0, 1]))

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

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$726$$ = $$2 \cdot 3 \cdot 11^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-1452$$ = $$-1 \cdot 2^{2} \cdot 3 \cdot 11^{2}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$3040$$ = $$2^{5} \cdot 5 \cdot 19$$ $$I_4$$ = $$-1107776$$ = $$-1 \cdot 2^{6} \cdot 19 \cdot 911$$ $$I_6$$ = $$-1033126976$$ = $$-1 \cdot 2^{6} \cdot 7^{3} \cdot 19 \cdot 2477$$ $$I_{10}$$ = $$-5947392$$ = $$-1 \cdot 2^{14} \cdot 3 \cdot 11^{2}$$ $$J_2$$ = $$380$$ = $$2^{2} \cdot 5 \cdot 19$$ $$J_4$$ = $$17556$$ = $$2^{2} \cdot 3 \cdot 7 \cdot 11 \cdot 19$$ $$J_6$$ = $$702601$$ = $$19 \cdot 36979$$ $$J_8$$ = $$-10306189$$ = $$-1 \cdot 19^{2} \cdot 28549$$ $$J_{10}$$ = $$-1452$$ = $$-1 \cdot 2^{2} \cdot 3 \cdot 11^{2}$$ $$g_1$$ = $$-1980879200000/363$$ $$g_2$$ = $$-7297976000/11$$ $$g_3$$ = $$-25363896100/363$$
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![0,-1,1],C![0,0,1],C![1,0,0]];

All rational points: (0 : -1 : 1), (0 : 0 : 1), (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: 2 (p = 2), 1 (p = 3), 1 (p = 11) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{10}\Z$$

### 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 66.c3
Elliptic curve 11.a3

### Endomorphisms

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

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ an order of index $$5$$ 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 $$\Q$$.