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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 22050z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.i1 | 22050z1 | \([1, -1, 0, -358542, -185734634]\) | \(-77626969/182250\) | \(-11967389094691406250\) | \([]\) | \(580608\) | \(2.3490\) | \(\Gamma_0(N)\)-optimal |
| 22050.i2 | 22050z2 | \([1, -1, 0, 3114333, 4110211741]\) | \(50872947671/140625000\) | \(-9234096523681640625000\) | \([]\) | \(1741824\) | \(2.8983\) |
Rank
sage: E.rank()
The elliptic curves in class 22050z have rank \(0\).
Complex multiplication
The elliptic curves in class 22050z do not have complex multiplication.Modular form 22050.2.a.z
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
