# Properties

 Label 22218.y Number of curves $6$ Conductor $22218$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("22218.y1")

sage: E.isogeny_class()

## Elliptic curves in class 22218.y

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22218.y1 22218t6 [1, 1, 1, -41861897, 104231607419] [2] 2162688
22218.y2 22218t4 [1, 1, 1, -9370717, -11044581181] [2] 1081344
22218.y3 22218t3 [1, 1, 1, -2684157, 1538915331] [2, 2] 1081344
22218.y4 22218t2 [1, 1, 1, -610477, -157354909] [2, 2] 540672
22218.y5 22218t1 [1, 1, 1, 66643, -13534621] [4] 270336 $$\Gamma_0(N)$$-optimal
22218.y6 22218t5 [1, 1, 1, 3314703, 7448992203] [2] 2162688

## Rank

sage: E.rank()

The elliptic curves in class 22218.y have rank $$1$$.

## Modular form 22218.2.a.y

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.