# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("69696.df1")

sage: E.isogeny_class()

## Elliptic curves in class 69696bn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
69696.df3 69696bn1 [0, 0, 0, -384780, 86653424]  737280 $$\Gamma_0(N)$$-optimal
69696.df4 69696bn2 [0, 0, 0, 312180, 365716208]  1474560
69696.df1 69696bn3 [0, 0, 0, -5611980, -5097474448]  2211840
69696.df2 69696bn4 [0, 0, 0, -2824140, -10163537296]  4423680

## Rank

sage: E.rank()

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

## Modular form 69696.2.a.df

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 