# Properties

 Label 22386.x Number of curves 4 Conductor 22386 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("22386.x1")

sage: E.isogeny_class()

## Elliptic curves in class 22386.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22386.x1 22386z4 [1, 0, 0, -1371509, -618338655] [2] 258048
22386.x2 22386z3 [1, 0, 0, -110709, -3584607] [2] 258048
22386.x3 22386z2 [1, 0, 0, -85749, -9659871] [2, 2] 129024
22386.x4 22386z1 [1, 0, 0, -3829, -239071] [2] 64512 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form 22386.2.a.x

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.