Properties

Label 22050.ej
Number of curves $8$
Conductor $22050$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("22050.ej1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 22050.ej

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22050.ej1 22050dz8 [1, -1, 1, -71122505, 145263350747] [2] 5308416  
22050.ej2 22050dz5 [1, -1, 1, -63515255, 194850052247] [2] 1769472  
22050.ej3 22050dz6 [1, -1, 1, -29778755, -60876586753] [2, 2] 2654208  
22050.ej4 22050dz3 [1, -1, 1, -29558255, -61846345753] [2] 1327104  
22050.ej5 22050dz2 [1, -1, 1, -3980255, 3028282247] [2, 2] 884736  
22050.ej6 22050dz4 [1, -1, 1, -893255, 7603216247] [2] 1769472  
22050.ej7 22050dz1 [1, -1, 1, -452255, -41077753] [2] 442368 \(\Gamma_0(N)\)-optimal
22050.ej8 22050dz7 [1, -1, 1, 8036995, -204954594253] [2] 5308416  

Rank

sage: E.rank()
 

The elliptic curves in class 22050.ej have rank \(0\).

Modular form 22050.2.a.ej

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} + q^{8} + 2q^{13} + q^{16} + 6q^{17} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.