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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 22050co
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22050.t1 | 22050co1 | \([1, -1, 0, -887742, 38976916]\) | \(461889917/263424\) | \(44126669254500000000\) | \([2]\) | \(737280\) | \(2.4593\) | \(\Gamma_0(N)\)-optimal |
22050.t2 | 22050co2 | \([1, -1, 0, 3522258, 307986916]\) | \(28849701763/16941456\) | \(-2837896416430031250000\) | \([2]\) | \(1474560\) | \(2.8058\) |
Rank
sage: E.rank()
The elliptic curves in class 22050co have rank \(0\).
Complex multiplication
The elliptic curves in class 22050co do not have complex multiplication.Modular form 22050.2.a.co
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.