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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 206310cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 206310.d3 | 206310cm1 | \([1, 1, 0, -292283, -56651427]\) | \(18653901818761/1469644800\) | \(217560174482227200\) | \([2]\) | \(2703360\) | \(2.0706\) | \(\Gamma_0(N)\)-optimal |
| 206310.d2 | 206310cm2 | \([1, 1, 0, -969403, 301003357]\) | \(680566353484681/128737440000\) | \(19057761377984160000\) | \([2, 2]\) | \(5406720\) | \(2.4172\) | |
| 206310.d1 | 206310cm3 | \([1, 1, 0, -14723403, 21737987757]\) | \(2384412229264108681/117869029200\) | \(17448846523188958800\) | \([2]\) | \(10813440\) | \(2.7637\) | |
| 206310.d4 | 206310cm4 | \([1, 1, 0, 1950677, 1767467533]\) | \(5545139013530999/12316931250000\) | \(-1823347867345631250000\) | \([2]\) | \(10813440\) | \(2.7637\) |
Rank
sage: E.rank()
The elliptic curves in class 206310cm have rank \(1\).
Complex multiplication
The elliptic curves in class 206310cm do not have complex multiplication.Modular form 206310.2.a.cm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
