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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 3630.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3630.f1 | 3630d5 | \([1, 1, 0, -20701287, 36244390329]\) | \(553808571467029327441/12529687500\) | \(22197105717187500\) | \([4]\) | \(184320\) | \(2.6607\) | |
| 3630.f2 | 3630d3 | \([1, 1, 0, -1430827, -657916211]\) | \(182864522286982801/463015182960\) | \(820259640539800560\) | \([2]\) | \(92160\) | \(2.3141\) | |
| 3630.f3 | 3630d4 | \([1, 1, 0, -1295307, 564555501]\) | \(135670761487282321/643043610000\) | \(1139190980775210000\) | \([2, 2]\) | \(92160\) | \(2.3141\) | |
| 3630.f4 | 3630d6 | \([1, 1, 0, -629807, 1144738401]\) | \(-15595206456730321/310672490129100\) | \(-550375267285598525100\) | \([2]\) | \(184320\) | \(2.6607\) | |
| 3630.f5 | 3630d2 | \([1, 1, 0, -124027, -1641251]\) | \(119102750067601/68309049600\) | \(121013648218425600\) | \([2, 2]\) | \(46080\) | \(1.9676\) | |
| 3630.f6 | 3630d1 | \([1, 1, 0, 30853, -185379]\) | \(1833318007919/1070530560\) | \(-1896510189404160\) | \([2]\) | \(23040\) | \(1.6210\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3630.f have rank \(1\).
Complex multiplication
The elliptic curves in class 3630.f do not have complex multiplication.Modular form 3630.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
