# Properties

 Label 221760.df Number of curves $2$ Conductor $221760$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("df1")

sage: E.isogeny_class()

## Elliptic curves in class 221760.df

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221760.df1 221760lu2 $$[0, 0, 0, -18588, -973712]$$ $$59466754384/121275$$ $$1448500838400$$ $$$$ $$491520$$ $$1.2199$$
221760.df2 221760lu1 $$[0, 0, 0, -768, -25688]$$ $$-67108864/343035$$ $$-256074255360$$ $$$$ $$245760$$ $$0.87338$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 221760.df have rank $$1$$.

## Complex multiplication

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

## Modular form 221760.2.a.df

sage: E.q_eigenform(10)

$$q - q^{5} - q^{7} + q^{11} + 6q^{13} - 2q^{17} + O(q^{20})$$ ## 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 LMFDB numbering.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 