# Properties

 Label 270720.b.270720.1 Conductor 270720 Discriminant -270720 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![1410, 0, 544, 0, 70, 0, 3], R![0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1410, 0, 544, 0, 70, 0, 3]), R([0, 1]))

$y^2 + xy = 3x^6 + 70x^4 + 544x^2 + 1410$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$270720$$ = $$2^{7} \cdot 3^{2} \cdot 5 \cdot 47$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-270720$$ = $$-1 \cdot 2^{7} \cdot 3^{2} \cdot 5 \cdot 47$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-25996160$$ = $$-1 \cdot 2^{7} \cdot 5 \cdot 151 \cdot 269$$ $$I_4$$ = $$3097408$$ = $$2^{6} \cdot 48397$$ $$I_6$$ = $$-26839668372480$$ = $$-1 \cdot 2^{10} \cdot 3^{2} \cdot 5 \cdot 17971 \cdot 32411$$ $$I_{10}$$ = $$-1108869120$$ = $$-1 \cdot 2^{19} \cdot 3^{2} \cdot 5 \cdot 47$$ $$J_2$$ = $$-3249520$$ = $$-1 \cdot 2^{4} \cdot 5 \cdot 151 \cdot 269$$ $$J_4$$ = $$439974144002$$ = $$2 \cdot 219987072001$$ $$J_6$$ = $$-79428031708101120$$ = $$-1 \cdot 2^{9} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 47 \cdot 6668079709$$ $$J_8$$ = $$16131432551454029721599$$ = $$16131432551454029721599$$ $$J_{10}$$ = $$-270720$$ = $$-1 \cdot 2^{7} \cdot 3^{2} \cdot 5 \cdot 47$$ $$g_1$$ = $$566129906277843097800186880000/423$$ $$g_2$$ = $$23588744365074445774311454400/423$$ $$g_3$$ = $$3098074718373623337958400$$
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 $\Q_{2}$ and $\Q_{7}$.

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: $$6$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 1.0 Real period: 3.3835115798391790478970284305 Tamagawa numbers: 1 (p = 2), 1 (p = 3), 1 (p = 5), 1 (p = 47) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

2-torsion field: splitting field of $$x^{8} - 4 x^{7} - 102 x^{6} + 320 x^{5} + 6277 x^{4} - 13092 x^{3} - 200796 x^{2} + 207396 x + 3390849$$ with Galois group $C_2^3$

### 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 192.c2
Elliptic curve 1410.j2

### 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 $$\Q$$.