# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("69696.bd1")

sage: E.isogeny_class()

## Elliptic curves in class 69696.bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
69696.bd1 69696cw4 [0, 0, 0, -10211916, 12548220784]  2949120
69696.bd2 69696cw2 [0, 0, 0, -802956, 86994160] [2, 2] 1474560
69696.bd3 69696cw1 [0, 0, 0, -454476, -116936336]  737280 $$\Gamma_0(N)$$-optimal
69696.bd4 69696cw3 [0, 0, 0, 3030324, 677319280]  2949120

## Rank

sage: E.rank()

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

## Modular form 69696.2.a.bd

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 