# Properties

 Label 69938e Number of curves 4 Conductor 69938 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("69938.f1")

sage: E.isogeny_class()

## Elliptic curves in class 69938e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
69938.f4 69938e1 [1, 1, 0, -105635, -8225587]  622080 $$\Gamma_0(N)$$-optimal
69938.f3 69938e2 [1, 1, 0, -1504395, -710682859]  1244160
69938.f2 69938e3 [1, 1, 0, -3602535, 2629975649]  1866240
69938.f1 69938e4 [1, 1, 0, -3952225, 2088165963]  3732480

## Rank

sage: E.rank()

The elliptic curves in class 69938e have rank $$0$$.

## Modular form 69938.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} + 2q^{3} + q^{4} - 2q^{6} - 4q^{7} - q^{8} + q^{9} + 2q^{12} - 2q^{13} + 4q^{14} + q^{16} - q^{18} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 