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SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 284592eu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 284592.eu4 | 284592eu1 | \([0, -1, 0, -3227352, -2229814992]\) | \(4354703137/1617\) | \(1380429777321627648\) | \([2]\) | \(7372800\) | \(2.4481\) | \(\Gamma_0(N)\)-optimal |
| 284592.eu3 | 284592eu2 | \([0, -1, 0, -3701672, -1530857040]\) | \(6570725617/2614689\) | \(2232154949929071906816\) | \([2, 2]\) | \(14745600\) | \(2.7947\) | |
| 284592.eu2 | 284592eu3 | \([0, -1, 0, -26943352, 52761707440]\) | \(2533811507137/58110129\) | \(49608504907607332786176\) | \([2, 2]\) | \(29491200\) | \(3.1413\) | |
| 284592.eu6 | 284592eu4 | \([0, -1, 0, 11950888, -11097701712]\) | \(221115865823/190238433\) | \(-162406182872112171159552\) | \([2]\) | \(29491200\) | \(3.1413\) | |
| 284592.eu1 | 284592eu5 | \([0, -1, 0, -428692392, 3416526069552]\) | \(10206027697760497/5557167\) | \(4744142736146628046848\) | \([2]\) | \(58982400\) | \(3.4878\) | |
| 284592.eu5 | 284592eu6 | \([0, -1, 0, 2938808, 163349605168]\) | \(3288008303/13504609503\) | \(-11528859053210777132101632\) | \([2]\) | \(58982400\) | \(3.4878\) |
Rank
sage: E.rank()
The elliptic curves in class 284592eu have rank \(0\).
Complex multiplication
The elliptic curves in class 284592eu do not have complex multiplication.Modular form 284592.2.a.eu
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
