# Properties

 Label 38440e Number of curves 4 Conductor 38440 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("38440.g1")

sage: E.isogeny_class()

## Elliptic curves in class 38440e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
38440.g3 38440e1 [0, 0, 0, -1922, -29791]  28800 $$\Gamma_0(N)$$-optimal
38440.g2 38440e2 [0, 0, 0, -6727, 178746] [2, 2] 57600
38440.g4 38440e3 [0, 0, 0, 12493, 1012894]  115200
38440.g1 38440e4 [0, 0, 0, -102827, 12690966]  115200

## Rank

sage: E.rank()

The elliptic curves in class 38440e have rank $$1$$.

## Modular form 38440.2.a.g

sage: E.q_eigenform(10)

$$q + q^{5} - 4q^{7} - 3q^{9} - 4q^{11} + 2q^{13} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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. 