Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("22050.bj1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 22050.bj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22050.bj1 22050be7 [1, -1, 0, -58802067, -173540248659] [2] 1327104  
22050.bj2 22050be8 [1, -1, 0, -5000067, -584458659] [2] 1327104  
22050.bj3 22050be6 [1, -1, 0, -3677067, -2707873659] [2, 2] 663552  
22050.bj4 22050be5 [1, -1, 0, -3180942, 2184414966] [2] 442368  
22050.bj5 22050be4 [1, -1, 0, -755442, -217491534] [2] 442368  
22050.bj6 22050be2 [1, -1, 0, -204192, 32224716] [2, 2] 221184  
22050.bj7 22050be3 [1, -1, 0, -149067, -72457659] [2] 331776  
22050.bj8 22050be1 [1, -1, 0, 16308, 2457216] [2] 110592 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Modular form 22050.2.a.bj

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.