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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 13650.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13650.d1 | 13650a7 | \([1, 1, 0, -1004761375, -12259064860625]\) | \(7179471593960193209684686321/49441793310\) | \(772528020468750\) | \([2]\) | \(2654208\) | \(3.3925\) | |
| 13650.d2 | 13650a6 | \([1, 1, 0, -62797625, -191567259375]\) | \(1752803993935029634719121/4599740941532100\) | \(71870952211439062500\) | \([2, 2]\) | \(1327104\) | \(3.0459\) | |
| 13650.d3 | 13650a8 | \([1, 1, 0, -62025875, -196504144125]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-1404710718224586855468750\) | \([2]\) | \(2654208\) | \(3.3925\) | |
| 13650.d4 | 13650a4 | \([1, 1, 0, -12410125, -16804746875]\) | \(13527956825588849127121/25701087819771000\) | \(401579497183921875000\) | \([2]\) | \(884736\) | \(2.8432\) | |
| 13650.d5 | 13650a3 | \([1, 1, 0, -3973125, -2917087875]\) | \(443915739051786565201/21894701746029840\) | \(342104714781716250000\) | \([2]\) | \(663552\) | \(2.6994\) | |
| 13650.d6 | 13650a2 | \([1, 1, 0, -1035125, -72121875]\) | \(7850236389974007121/4400862921000000\) | \(68763483140625000000\) | \([2, 2]\) | \(442368\) | \(2.4966\) | |
| 13650.d7 | 13650a1 | \([1, 1, 0, -643125, 197182125]\) | \(1882742462388824401/11650189824000\) | \(182034216000000000\) | \([2]\) | \(221184\) | \(2.1500\) | \(\Gamma_0(N)\)-optimal |
| 13650.d8 | 13650a5 | \([1, 1, 0, 4067875, -567112875]\) | \(476437916651992691759/284661685546875000\) | \(-4447838836669921875000\) | \([2]\) | \(884736\) | \(2.8432\) |
Rank
sage: E.rank()
The elliptic curves in class 13650.d have rank \(1\).
Complex multiplication
The elliptic curves in class 13650.d do not have complex multiplication.Modular form 13650.2.a.d
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
