Properties

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

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

Elliptic curves in class 22218z

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

Rank

sage: E.rank()
 

The elliptic curves in class 22218z 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.