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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 3549c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3549.c6 | 3549c1 | \([1, 0, 1, 165, -167]\) | \(103823/63\) | \(-304088967\) | \([2]\) | \(1152\) | \(0.31696\) | \(\Gamma_0(N)\)-optimal |
| 3549.c5 | 3549c2 | \([1, 0, 1, -680, -1519]\) | \(7189057/3969\) | \(19157604921\) | \([2, 2]\) | \(2304\) | \(0.66353\) | |
| 3549.c2 | 3549c3 | \([1, 0, 1, -8285, -290509]\) | \(13027640977/21609\) | \(104302515681\) | \([2, 2]\) | \(4608\) | \(1.0101\) | |
| 3549.c3 | 3549c4 | \([1, 0, 1, -6595, 204323]\) | \(6570725617/45927\) | \(221680856943\) | \([2]\) | \(4608\) | \(1.0101\) | |
| 3549.c1 | 3549c5 | \([1, 0, 1, -132500, -18574957]\) | \(53297461115137/147\) | \(709540923\) | \([2]\) | \(9216\) | \(1.3567\) | |
| 3549.c4 | 3549c6 | \([1, 0, 1, -5750, -471001]\) | \(-4354703137/17294403\) | \(-83476780050027\) | \([2]\) | \(9216\) | \(1.3567\) |
Rank
sage: E.rank()
The elliptic curves in class 3549c have rank \(1\).
Complex multiplication
The elliptic curves in class 3549c do not have complex multiplication.Modular form 3549.2.a.c
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.
