# Properties

 Label 15360.f.983040.2 Conductor 15360 Discriminant -983040 Sato-Tate group $N(G_{1,3})$ $\End(J_{\overline{\Q}}) \otimes \R$ $\C \times \R$ $\overline{\Q}$-simple no $\mathrm{GL}_2$-type yes

# Related objects

Show commands for: Magma / SageMath

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-30, 0, -37, 0, -15, 0, -2], R![]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-30, 0, -37, 0, -15, 0, -2]), R([]))

$y^2 = -2x^6 - 15x^4 - 37x^2 - 30$

## Invariants

 magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(15360,2),R![1]>*])); Factorization($1); $N$ = $15360$ = $2^{10} \cdot 3 \cdot 5$ magma: Discriminant(C); Factorization(Integers()!$1); $\Delta$ = $-983040$ = $-1 \cdot 2^{16} \cdot 3 \cdot 5$

### G2 invariants

magma: G2Invariants(C);

 $I_2$ = $-372480$ = $-1 \cdot 2^{8} \cdot 3 \cdot 5 \cdot 97$ $I_4$ = $918528$ = $2^{10} \cdot 3 \cdot 13 \cdot 23$ $I_6$ = $-114034114560$ = $-1 \cdot 2^{15} \cdot 3 \cdot 5 \cdot 232003$ $I_{10}$ = $-4026531840$ = $-1 \cdot 2^{28} \cdot 3 \cdot 5$ $J_2$ = $-46560$ = $-1 \cdot 2^{5} \cdot 3 \cdot 5 \cdot 97$ $J_4$ = $90316832$ = $2^{5} \cdot 113 \cdot 24977$ $J_6$ = $-233570058240$ = $-1 \cdot 2^{14} \cdot 3 \cdot 5 \cdot 19 \cdot 50021$ $J_8$ = $679472942284544$ = $2^{8} \cdot 53 \cdot 20921 \cdot 2393723$ $J_{10}$ = $-983040$ = $-1 \cdot 2^{16} \cdot 3 \cdot 5$ $g_1$ = $222583859461440000$ $g_2$ = $9273345076342800$ $g_3$ = $515076721401600$
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; HasPointsLocallyEverywhere(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);

magma: [];

### There are no rational points

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

### Number of rational Weierstrass points:

$0$

## Invariants of the Jacobian:

### Analytic rank:

$1$

### Torsion:

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

### 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 480.b3
Elliptic curve 32.a3

### 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 $K = \Q (a) \simeq$ $\Q(\sqrt{-1})$ with defining polynomial $x^{2} + 1$

not of $\GL_2$-type over $\overline{\Q}$

Endomorphism ring over $\overline{\Q}$:
 $\End (J_{\overline{\Q}})$ $\simeq$ an order of index $4$ in $\Z \times \Z [\sqrt{-1}]$ $\End (J_{\overline{\Q}}) \otimes \Q$ $\simeq$ $\Q$ $\times$ $\Q(\sqrt{-1})$ $\End (J_{\overline{\Q}}) \otimes \R$ $\simeq$ $\R \times \C$