# Properties

 Label 2916.a.139968.1 Conductor 2916 Discriminant -139968 Sato-Tate group $J(E_1)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-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, 0, -6, 5, -3, 1], R![1, 1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, -6, 5, -3, 1]), R([1, 1, 1]))

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

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$2916$$ = $$2^{2} \cdot 3^{6}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-139968$$ = $$-1 \cdot 2^{6} \cdot 3^{7}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-1944$$ = $$-1 \cdot 2^{3} \cdot 3^{5}$$ $$I_4$$ = $$453924$$ = $$2^{2} \cdot 3^{5} \cdot 467$$ $$I_6$$ = $$-384120792$$ = $$-1 \cdot 2^{3} \cdot 3^{5} \cdot 11^{2} \cdot 23 \cdot 71$$ $$I_{10}$$ = $$-573308928$$ = $$-1 \cdot 2^{18} \cdot 3^{7}$$ $$J_2$$ = $$-243$$ = $$-1 \cdot 3^{5}$$ $$J_4$$ = $$-2268$$ = $$-1 \cdot 2^{2} \cdot 3^{4} \cdot 7$$ $$J_6$$ = $$314496$$ = $$2^{7} \cdot 3^{3} \cdot 7 \cdot 13$$ $$J_8$$ = $$-20391588$$ = $$-1 \cdot 2^{2} \cdot 3^{9} \cdot 7 \cdot 37$$ $$J_{10}$$ = $$-139968$$ = $$-1 \cdot 2^{6} \cdot 3^{7}$$ $$g_1$$ = $$387420489/64$$ $$g_2$$ = $$-3720087/16$$ $$g_3$$ = $$-132678$$
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![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,1,0]];

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

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: $$0$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 1.0 Real period: 19.520681040807033229434261445 Tamagawa numbers: 9 (p = 2), 3 (p = 3) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{3}\Z \times \Z/{9}\Z$$

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $J(E_1)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

### Decomposition

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 54.b2
Elliptic curve 54.a2

### 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 $$K = \Q (a) \simeq$$ $$\Q(\sqrt{-3})$$ 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 Eichler order of index $$9$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$