Properties

Label 223440.bf
Number of curves $4$
Conductor $223440$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bf1")
 
E.isogeny_class()
 

Elliptic curves in class 223440.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
223440.bf1 223440hk3 \([0, -1, 0, -1489616, 700273680]\) \(3034301922374404/1425\) \(171673420800\) \([2]\) \(1572864\) \(1.9294\)  
223440.bf2 223440hk4 \([0, -1, 0, -111736, 6283936]\) \(1280615525284/601171875\) \(72424724400000000\) \([2]\) \(1572864\) \(1.9294\)  
223440.bf3 223440hk2 \([0, -1, 0, -93116, 10961280]\) \(2964647793616/2030625\) \(61158656160000\) \([2, 2]\) \(786432\) \(1.5828\)  
223440.bf4 223440hk1 \([0, -1, 0, -4671, 241746]\) \(-5988775936/9774075\) \(-18398562394800\) \([2]\) \(393216\) \(1.2363\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 223440.bf have rank \(1\).

Complex multiplication

The elliptic curves in class 223440.bf do not have complex multiplication.

Modular form 223440.2.a.bf

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} + 2 q^{13} + q^{15} + 6 q^{17} + q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.