# Properties

 Label 22386s Number of curves 4 Conductor 22386 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("22386.o1")

sage: E.isogeny_class()

## Elliptic curves in class 22386s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22386.o3 22386s1 [1, 1, 1, -2104, -38023]  15360 $$\Gamma_0(N)$$-optimal
22386.o2 22386s2 [1, 1, 1, -2184, -35079] [2, 2] 30720
22386.o4 22386s3 [1, 1, 1, 3556, -182023]  61440
22386.o1 22386s4 [1, 1, 1, -9204, 301881]  61440

## Rank

sage: E.rank()

The elliptic curves in class 22386s have rank $$1$$.

## Modular form 22386.2.a.o

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 