# Properties

 Label 221760es Number of curves $4$ Conductor $221760$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("es1")

sage: E.isogeny_class()

## Elliptic curves in class 221760es

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221760.cl3 221760es1 $$[0, 0, 0, -95808, 11583448]$$ $$-130287139815424/2250652635$$ $$-1680103189416960$$ $$[2]$$ $$1327104$$ $$1.7194$$ $$\Gamma_0(N)$$-optimal
221760.cl2 221760es2 $$[0, 0, 0, -1539228, 735025552]$$ $$33766427105425744/9823275$$ $$117328567910400$$ $$[2]$$ $$2654208$$ $$2.0660$$
221760.cl4 221760es3 $$[0, 0, 0, 370752, 55510072]$$ $$7549996227362816/6152409907875$$ $$-4592749386589056000$$ $$[2]$$ $$3981312$$ $$2.2687$$
221760.cl1 221760es4 $$[0, 0, 0, -1785468, 484166608]$$ $$52702650535889104/22020583921875$$ $$263012445045504000000$$ $$[2]$$ $$7962624$$ $$2.6153$$

## Rank

sage: E.rank()

The elliptic curves in class 221760es have rank $$0$$.

## Complex multiplication

The elliptic curves in class 221760es do not have complex multiplication.

## Modular form 221760.2.a.es

sage: E.q_eigenform(10)

$$q - q^{5} - q^{7} + q^{11} - 2q^{13} + 6q^{17} + 8q^{19} + O(q^{20})$$

## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels.