# Properties

 Label 55440en Number of curves 4 Conductor 55440 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("55440.eh1")

sage: E.isogeny_class()

## Elliptic curves in class 55440en

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55440.eh3 55440en1 [0, 0, 0, -72147, -6476654] [2] 294912 $$\Gamma_0(N)$$-optimal
55440.eh2 55440en2 [0, 0, 0, -305427, 58515154] [2, 2] 589824
55440.eh4 55440en3 [0, 0, 0, 407373, 291030514] [2] 1179648
55440.eh1 55440en4 [0, 0, 0, -4750707, 3985475506] [2] 1179648

## Rank

sage: E.rank()

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

## Modular form 55440.2.a.eh

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.