# Properties

 Label 69696cx Number of curves 4 Conductor 69696 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("69696.bf1")

sage: E.isogeny_class()

## Elliptic curves in class 69696cx

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
69696.bf4 69696cx1 [0, 0, 0, 2904, 1650440]  368640 $$\Gamma_0(N)$$-optimal
69696.bf3 69696cx2 [0, 0, 0, -193116, 31837520] [2, 2] 737280
69696.bf2 69696cx3 [0, 0, 0, -454476, -72811024]  1474560
69696.bf1 69696cx4 [0, 0, 0, -3068076, 2068459184]  1474560

## Rank

sage: E.rank()

The elliptic curves in class 69696cx have rank $$1$$.

## Modular form 69696.2.a.bf

sage: E.q_eigenform(10)

$$q - 2q^{5} - 4q^{7} + 6q^{13} + 6q^{17} - 8q^{19} + O(q^{20})$$

## 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. 