# Properties

 Label 221760lz Number of curves 4 Conductor 221760 CM no Rank 2 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("221760.p1")

sage: E.isogeny_class()

## Elliptic curves in class 221760lz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
221760.p3 221760lz1 [0, 0, 0, -288588, 51813232] [2] 2359296 $$\Gamma_0(N)$$-optimal
221760.p2 221760lz2 [0, 0, 0, -1221708, -468121232] [2, 2] 4718592
221760.p4 221760lz3 [0, 0, 0, 1629492, -2328244112] [2] 9437184
221760.p1 221760lz4 [0, 0, 0, -19002828, -31883804048] [2] 9437184

## Rank

sage: E.rank()

The elliptic curves in class 221760lz have rank $$2$$.

## Modular form 221760.2.a.p

sage: E.q_eigenform(10)

$$q - q^{5} - q^{7} - q^{11} - 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.