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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 22050bp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.s1 | 22050bp1 | \([1, -1, 0, -7317, 543591]\) | \(-77626969/182250\) | \(-101721128906250\) | \([]\) | \(82944\) | \(1.3761\) | \(\Gamma_0(N)\)-optimal |
| 22050.s2 | 22050bp2 | \([1, -1, 0, 63558, -12001284]\) | \(50872947671/140625000\) | \(-78488525390625000\) | \([]\) | \(248832\) | \(1.9254\) |
Rank
sage: E.rank()
The elliptic curves in class 22050bp have rank \(1\).
Complex multiplication
The elliptic curves in class 22050bp do not have complex multiplication.Modular form 22050.2.a.bp
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.
