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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 16650.cb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16650.cb1 | 16650bu3 | \([1, -1, 1, -76726130, -258660935503]\) | \(4385367890843575421521/24975000000\) | \(284480859375000000\) | \([2]\) | \(1327104\) | \(2.9608\) | |
| 16650.cb2 | 16650bu5 | \([1, -1, 1, -68203130, 215872164497]\) | \(3080272010107543650001/15465841417699560\) | \(176165599898484050625000\) | \([2]\) | \(2654208\) | \(3.3074\) | |
| 16650.cb3 | 16650bu4 | \([1, -1, 1, -6598130, -731015503]\) | \(2788936974993502801/1593609593601600\) | \(18152209277118225000000\) | \([2, 2]\) | \(1327104\) | \(2.9608\) | |
| 16650.cb4 | 16650bu2 | \([1, -1, 1, -4798130, -4035815503]\) | \(1072487167529950801/2554882560000\) | \(29101709160000000000\) | \([2, 2]\) | \(663552\) | \(2.6143\) | |
| 16650.cb5 | 16650bu1 | \([1, -1, 1, -190130, -109799503]\) | \(-66730743078481/419010969600\) | \(-4772796825600000000\) | \([2]\) | \(331776\) | \(2.2677\) | \(\Gamma_0(N)\)-optimal |
| 16650.cb6 | 16650bu6 | \([1, -1, 1, 26206870, -5848595503]\) | \(174751791402194852399/102423900876336360\) | \(-1166672245919518850625000\) | \([2]\) | \(2654208\) | \(3.3074\) |
Rank
sage: E.rank()
The elliptic curves in class 16650.cb have rank \(1\).
Complex multiplication
The elliptic curves in class 16650.cb do not have complex multiplication.Modular form 16650.2.a.cb
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
