# Properties

 Label 55440.cf Number of curves $6$ Conductor $55440$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("55440.cf1")

sage: E.isogeny_class()

## Elliptic curves in class 55440.cf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55440.cf1 55440dr6 [0, 0, 0, -1908003, -1014368798]  1048576
55440.cf2 55440dr4 [0, 0, 0, -126003, -13953998] [2, 2] 524288
55440.cf3 55440dr2 [0, 0, 0, -38883, 2755618] [2, 2] 262144
55440.cf4 55440dr1 [0, 0, 0, -38163, 2869522]  131072 $$\Gamma_0(N)$$-optimal
55440.cf5 55440dr3 [0, 0, 0, 36717, 12175378]  524288
55440.cf6 55440dr5 [0, 0, 0, 262077, -82954622]  1048576

## Rank

sage: E.rank()

The elliptic curves in class 55440.cf have rank $$1$$.

## Modular form 55440.2.a.cf

sage: E.q_eigenform(10)

$$q - q^{5} + q^{7} + q^{11} - 2q^{13} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 