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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 6045.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6045.i1 | 6045g3 | \([1, 0, 1, -53734, 4789721]\) | \(17157721205076026329/816075\) | \(816075\) | \([4]\) | \(10240\) | \(1.0601\) | |
| 6045.i2 | 6045g2 | \([1, 0, 1, -3359, 74621]\) | \(4189554574052329/913550625\) | \(913550625\) | \([2, 2]\) | \(5120\) | \(0.71348\) | |
| 6045.i3 | 6045g4 | \([1, 0, 1, -2984, 92021]\) | \(-2937047271278329/1978251246075\) | \(-1978251246075\) | \([2]\) | \(10240\) | \(1.0601\) | |
| 6045.i4 | 6045g1 | \([1, 0, 1, -234, 871]\) | \(1408317602329/472265625\) | \(472265625\) | \([2]\) | \(2560\) | \(0.36691\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6045.i have rank \(1\).
Complex multiplication
The elliptic curves in class 6045.i do not have complex multiplication.Modular form 6045.2.a.i
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 LMFDB 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 LMFDB labels.
