# Properties

 Label 221760.es Number of curves 4 Conductor 221760 CM no Rank 1 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("221760.es1")

sage: E.isogeny_class()

## Elliptic curves in class 221760.es

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
221760.es1 221760lg4 [0, 0, 0, -15042828, 21820184048] [2] 15925248
221760.es2 221760lg2 [0, 0, 0, -2059788, -1127734288] [2] 5308416
221760.es3 221760lg1 [0, 0, 0, -32268, -43416592] [2] 2654208 $$\Gamma_0(N)$$-optimal
221760.es4 221760lg3 [0, 0, 0, 290292, 1169538032] [2] 7962624

## Rank

sage: E.rank()

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

## Modular form 221760.2.a.es

sage: E.q_eigenform(10)

$$q - q^{5} + q^{7} - q^{11} + 4q^{13} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.