Properties

Label 22050er
Number of curves $6$
Conductor $22050$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("22050.df1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 22050er

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22050.df6 22050er1 [1, -1, 1, 110020, -19684353] [2] 294912 \(\Gamma_0(N)\)-optimal
22050.df5 22050er2 [1, -1, 1, -771980, -203140353] [2, 2] 589824  
22050.df4 22050er3 [1, -1, 1, -4079480, 2998519647] [2] 1179648  
22050.df2 22050er4 [1, -1, 1, -11576480, -15156568353] [2, 2] 1179648  
22050.df3 22050er5 [1, -1, 1, -10804730, -17264989353] [2] 2359296  
22050.df1 22050er6 [1, -1, 1, -185220230, -970197193353] [2] 2359296  

Rank

sage: E.rank()
 

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

Modular form 22050.2.a.df

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} + q^{8} - 4q^{11} - 2q^{13} + q^{16} - 2q^{17} + 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.