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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 169050bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169050.ic2 | 169050bk1 | \([1, 0, 0, -1975338, -1068722208]\) | \(463702796512201/15214500\) | \(27968292351562500\) | \([2]\) | \(3317760\) | \(2.2495\) | \(\Gamma_0(N)\)-optimal |
| 169050.ic3 | 169050bk2 | \([1, 0, 0, -1889588, -1165705458]\) | \(-405897921250921/84358968750\) | \(-155074192413574218750\) | \([2]\) | \(6635520\) | \(2.5961\) | |
| 169050.ic1 | 169050bk3 | \([1, 0, 0, -3537213, 841818417]\) | \(2662558086295801/1374177967680\) | \(2526104120618505000000\) | \([2]\) | \(9953280\) | \(2.7988\) | |
| 169050.ic4 | 169050bk4 | \([1, 0, 0, 13269787, 6539391417]\) | \(140574743422291079/91397357868600\) | \(-168012621185670646875000\) | \([2]\) | \(19906560\) | \(3.1454\) |
Rank
sage: E.rank()
The elliptic curves in class 169050bk have rank \(1\).
Complex multiplication
The elliptic curves in class 169050bk do not have complex multiplication.Modular form 169050.2.a.bk
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
