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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 360789m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 360789.m5 | 360789m1 | \([1, 1, 0, -20201, 1576896]\) | \(-1532808577/938223\) | \(-558076920698583\) | \([2]\) | \(1433600\) | \(1.5316\) | \(\Gamma_0(N)\)-optimal |
| 360789.m4 | 360789m2 | \([1, 1, 0, -360806, 83253975]\) | \(8732907467857/1656369\) | \(985246909381449\) | \([2, 2]\) | \(2867200\) | \(1.8782\) | |
| 360789.m1 | 360789m3 | \([1, 1, 0, -5772641, 5335981026]\) | \(35765103905346817/1287\) | \(765537614127\) | \([2]\) | \(5734400\) | \(2.2247\) | |
| 360789.m3 | 360789m4 | \([1, 1, 0, -398651, 64672080]\) | \(11779205551777/3763454409\) | \(2238590449993472289\) | \([2, 2]\) | \(5734400\) | \(2.2247\) | |
| 360789.m6 | 360789m5 | \([1, 1, 0, 1127764, 442307151]\) | \(266679605718863/296110251723\) | \(-176133283312020832083\) | \([2]\) | \(11468800\) | \(2.5713\) | |
| 360789.m2 | 360789m6 | \([1, 1, 0, -2530586, -1501447371]\) | \(3013001140430737/108679952667\) | \(64645370371507747107\) | \([2]\) | \(11468800\) | \(2.5713\) |
Rank
sage: E.rank()
The elliptic curves in class 360789m have rank \(1\).
Complex multiplication
The elliptic curves in class 360789m do not have complex multiplication.Modular form 360789.2.a.m
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
