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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 206910.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 206910.cm1 | 206910dk2 | \([1, -1, 0, -2176389, -1232646827]\) | \(882774443450089/2166000000\) | \(2797319620854000000\) | \([2]\) | \(6881280\) | \(2.4178\) | |
| 206910.cm2 | 206910dk1 | \([1, -1, 0, -85509, -33736235]\) | \(-53540005609/350208000\) | \(-452282414487552000\) | \([2]\) | \(3440640\) | \(2.0712\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 206910.cm have rank \(0\).
Complex multiplication
The elliptic curves in class 206910.cm do not have complex multiplication.Modular form 206910.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
