Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("22050.fn1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 22050eq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22050.fn5 22050eq1 [1, -1, 1, -44330, 7944297] [2] 196608 \(\Gamma_0(N)\)-optimal
22050.fn4 22050eq2 [1, -1, 1, -926330, 343104297] [2, 2] 393216  
22050.fn3 22050eq3 [1, -1, 1, -1146830, 167586297] [2, 2] 786432  
22050.fn1 22050eq4 [1, -1, 1, -14817830, 21958278297] [2] 786432  
22050.fn6 22050eq5 [1, -1, 1, 4255420, 1291254297] [2] 1572864  
22050.fn2 22050eq6 [1, -1, 1, -10077080, -12191879703] [2] 1572864  

Rank

sage: E.rank()
 

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

Modular form 22050.2.a.fn

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} + q^{8} + 4q^{11} + 6q^{13} + q^{16} - 2q^{17} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.