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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 4650.bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4650.bp1 | 4650bi3 | \([1, 0, 0, -165713, 25950417]\) | \(32208729120020809/658986840\) | \(10296669375000\) | \([2]\) | \(27648\) | \(1.6164\) | |
| 4650.bp2 | 4650bi2 | \([1, 0, 0, -10713, 375417]\) | \(8702409880009/1120910400\) | \(17514225000000\) | \([2, 2]\) | \(13824\) | \(1.2699\) | |
| 4650.bp3 | 4650bi1 | \([1, 0, 0, -2713, -48583]\) | \(141339344329/17141760\) | \(267840000000\) | \([2]\) | \(6912\) | \(0.92330\) | \(\Gamma_0(N)\)-optimal |
| 4650.bp4 | 4650bi4 | \([1, 0, 0, 16287, 1968417]\) | \(30579142915511/124675335000\) | \(-1948052109375000\) | \([2]\) | \(27648\) | \(1.6164\) |
Rank
sage: E.rank()
The elliptic curves in class 4650.bp have rank \(1\).
Complex multiplication
The elliptic curves in class 4650.bp do not have complex multiplication.Modular form 4650.2.a.bp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
