# Properties

 Label 390390bf Number of curves 4 Conductor 390390 CM no Rank 2 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("390390.bf1")

sage: E.isogeny_class()

## Elliptic curves in class 390390bf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
390390.bf3 390390bf1 [1, 1, 0, -84672, -8269776]  3538944 $$\Gamma_0(N)$$-optimal
390390.bf2 390390bf2 [1, 1, 0, -358452, 74247516] [2, 2] 7077888
390390.bf1 390390bf3 [1, 1, 0, -5575482, 5064858414]  14155776
390390.bf4 390390bf4 [1, 1, 0, 478098, 370218906]  14155776

## Rank

sage: E.rank()

The elliptic curves in class 390390bf have rank $$2$$.

## Modular form 390390.2.a.bf

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - q^{10} + q^{11} - q^{12} - q^{14} - q^{15} + q^{16} + 2q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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. 