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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 22050bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.cf3 | 22050bm1 | \([1, -1, 0, -1332, 22896]\) | \(-121945/32\) | \(-68612896800\) | \([]\) | \(21600\) | \(0.79616\) | \(\Gamma_0(N)\)-optimal |
| 22050.cf4 | 22050bm2 | \([1, -1, 0, 9693, -168939]\) | \(46969655/32768\) | \(-70259606323200\) | \([]\) | \(64800\) | \(1.3455\) | |
| 22050.cf2 | 22050bm3 | \([1, -1, 0, -5742, -1974834]\) | \(-25/2\) | \(-1675119550781250\) | \([]\) | \(108000\) | \(1.6009\) | |
| 22050.cf1 | 22050bm4 | \([1, -1, 0, -1383867, -626265459]\) | \(-349938025/8\) | \(-6700478203125000\) | \([]\) | \(324000\) | \(2.1502\) |
Rank
sage: E.rank()
The elliptic curves in class 22050bm have rank \(1\).
Complex multiplication
The elliptic curves in class 22050bm do not have complex multiplication.Modular form 22050.2.a.bm
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 5 & 15 \\ 3 & 1 & 15 & 5 \\ 5 & 15 & 1 & 3 \\ 15 & 5 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
