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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 99450bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 99450.bv3 | 99450bf1 | \([1, -1, 0, -132192, -18440784]\) | \(22428153804601/35802000\) | \(407807156250000\) | \([2]\) | \(1032192\) | \(1.7011\) | \(\Gamma_0(N)\)-optimal |
| 99450.bv2 | 99450bf2 | \([1, -1, 0, -172692, -6169284]\) | \(50002789171321/27473062500\) | \(312935352539062500\) | \([2, 2]\) | \(2064384\) | \(2.0477\) | |
| 99450.bv4 | 99450bf3 | \([1, -1, 0, 671058, -49200534]\) | \(2933972022568679/1789082460750\) | \(-20378767404480468750\) | \([2]\) | \(4128768\) | \(2.3943\) | |
| 99450.bv1 | 99450bf4 | \([1, -1, 0, -1664442, 821751966]\) | \(44769506062996441/323730468750\) | \(3687492370605468750\) | \([2]\) | \(4128768\) | \(2.3943\) |
Rank
sage: E.rank()
The elliptic curves in class 99450bf have rank \(1\).
Complex multiplication
The elliptic curves in class 99450bf do not have complex multiplication.Modular form 99450.2.a.bf
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.
