# Properties

 Label 40a Number of curves 4 Conductor 40 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("40.a1")

sage: E.isogeny_class()

## Elliptic curves in class 40a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
40.a2 40a1 [0, 0, 0, -7, -6] [2, 2] 2 $$\Gamma_0(N)$$-optimal
40.a1 40a2 [0, 0, 0, -107, -426]  4
40.a3 40a3 [0, 0, 0, -2, 1]  4
40.a4 40a4 [0, 0, 0, 13, -34]  4

## Rank

sage: E.rank()

The elliptic curves in class 40a have rank $$0$$.

## Modular form40.2.a.a

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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 