# Properties

 Label 22218.z Number of curves $6$ Conductor $22218$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("22218.z1")

sage: E.isogeny_class()

## Elliptic curves in class 22218.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22218.z1 22218z4 [1, 1, 1, -710987, -231045667] [2] 180224
22218.z2 22218z6 [1, 1, 1, -483517, 127965473] [2] 360448
22218.z3 22218z3 [1, 1, 1, -55027, -1781299] [2, 2] 180224
22218.z4 22218z2 [1, 1, 1, -44447, -3622219] [2, 2] 90112
22218.z5 22218z1 [1, 1, 1, -2127, -84267] [2] 45056 $$\Gamma_0(N)$$-optimal
22218.z6 22218z5 [1, 1, 1, 204183, -13497591] [2] 360448

## Rank

sage: E.rank()

The elliptic curves in class 22218.z have rank $$0$$.

## Modular form 22218.2.a.z

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.