Properties

Label 2205k
Number of curves 4
Conductor 2205
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("2205.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2205k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2205.b3 2205k1 [1, -1, 1, -1112, 13794] [4] 1536 \(\Gamma_0(N)\)-optimal
2205.b2 2205k2 [1, -1, 1, -3317, -55884] [2, 2] 3072  
2205.b1 2205k3 [1, -1, 1, -49622, -4241856] [2] 6144  
2205.b4 2205k4 [1, -1, 1, 7708, -355764] [2] 6144  

Rank

sage: E.rank()
 

The elliptic curves in class 2205k have rank \(0\).

Modular form 2205.2.a.b

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{4} + q^{5} + 3q^{8} - q^{10} + 6q^{13} - q^{16} + 2q^{17} + 8q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.