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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)
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.