# Properties

 Label 68354i Number of curves 2 Conductor 68354 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("68354.f1")

sage: E.isogeny_class()

## Elliptic curves in class 68354i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
68354.f2 68354i1 [1, -1, 1, 621237, -226146597]  3040352 $$\Gamma_0(N)$$-optimal
68354.f1 68354i2 [1, -1, 1, -550339173, -4969145682177] [] 21282464

## Rank

sage: E.rank()

The elliptic curves in class 68354i have rank $$1$$.

## Modular form 68354.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} - 3q^{3} + q^{4} - q^{5} - 3q^{6} + q^{7} + q^{8} + 6q^{9} - q^{10} + q^{11} - 3q^{12} + q^{13} + q^{14} + 3q^{15} + q^{16} + 4q^{17} + 6q^{18} + 6q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 