# Properties

 Label 39.a Number of curves 4 Conductor 39 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("39.a1")

sage: E.isogeny_class()

## Elliptic curves in class 39.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
39.a1 39a2 [1, 1, 0, -69, -252]  4
39.a2 39a3 [1, 1, 0, -19, 22]  4
39.a3 39a1 [1, 1, 0, -4, -5] [2, 2] 2 $$\Gamma_0(N)$$-optimal
39.a4 39a4 [1, 1, 0, 1, 0]  4

## Rank

sage: E.rank()

The elliptic curves in class 39.a have rank $$0$$.

## Modular form39.2.a.a

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} - 4q^{7} - 3q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} + q^{13} - 4q^{14} - 2q^{15} - q^{16} + 2q^{17} + q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 