Properties

Label 22050.bo
Number of curves 8
Conductor 22050
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("22050.bo1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 22050.bo

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22050.bo1 22050bd8 [1, -1, 0, -3872338542, -92747787595884] [2] 10616832  
22050.bo2 22050bd6 [1, -1, 0, -242026542, -1449071107884] [2, 2] 5308416  
22050.bo3 22050bd7 [1, -1, 0, -224386542, -1669271227884] [2] 10616832  
22050.bo4 22050bd5 [1, -1, 0, -48041667, -125901267759] [2] 3538944  
22050.bo5 22050bd3 [1, -1, 0, -16234542, -19130371884] [2] 2654208  
22050.bo6 22050bd2 [1, -1, 0, -6367167, 3248007741] [2, 2] 1769472  
22050.bo7 22050bd1 [1, -1, 0, -5485167, 4944093741] [2] 884736 \(\Gamma_0(N)\)-optimal
22050.bo8 22050bd4 [1, -1, 0, 21195333, 23837195241] [2] 3538944  

Rank

sage: E.rank()
 

The elliptic curves in class 22050.bo have rank \(1\).

Modular form 22050.2.a.bo

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.