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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1122.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1122.e1 | 1122e5 | \([1, 1, 1, -394944, -95697123]\) | \(6812873765474836663297/74052\) | \(74052\) | \([2]\) | \(4096\) | \(1.4379\) | |
| 1122.e2 | 1122e3 | \([1, 1, 1, -24684, -1502979]\) | \(1663303207415737537/5483698704\) | \(5483698704\) | \([2, 2]\) | \(2048\) | \(1.0914\) | |
| 1122.e3 | 1122e6 | \([1, 1, 1, -24344, -1545955]\) | \(-1595514095015181697/95635786040388\) | \(-95635786040388\) | \([2]\) | \(4096\) | \(1.4379\) | |
| 1122.e4 | 1122e2 | \([1, 1, 1, -1564, -23299]\) | \(423108074414017/23284318464\) | \(23284318464\) | \([2, 4]\) | \(1024\) | \(0.74478\) | |
| 1122.e5 | 1122e1 | \([1, 1, 1, -284, 1277]\) | \(2533811507137/625016832\) | \(625016832\) | \([4]\) | \(512\) | \(0.39821\) | \(\Gamma_0(N)\)-optimal |
| 1122.e6 | 1122e4 | \([1, 1, 1, 1076, -90883]\) | \(137763859017023/3683199928848\) | \(-3683199928848\) | \([4]\) | \(2048\) | \(1.0914\) |
Rank
sage: E.rank()
The elliptic curves in class 1122.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1122.e do not have complex multiplication.Modular form 1122.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
