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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 17640cb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17640.p3 | 17640cb1 | \([0, 0, 0, -12344178, -16690170427]\) | \(151591373397612544/32558203125\) | \(44678252620181250000\) | \([2]\) | \(737280\) | \(2.7654\) | \(\Gamma_0(N)\)-optimal |
| 17640.p2 | 17640cb2 | \([0, 0, 0, -13722303, -12733022302]\) | \(13015144447800784/4341909875625\) | \(95331524547570867360000\) | \([2, 2]\) | \(1474560\) | \(3.1119\) | |
| 17640.p1 | 17640cb3 | \([0, 0, 0, -89353803, 315613571798]\) | \(898353183174324196/29899176238575\) | \(2625880439887819077196800\) | \([2]\) | \(2949120\) | \(3.4585\) | |
| 17640.p4 | 17640cb4 | \([0, 0, 0, 39859197, -87822136402]\) | \(79743193254623804/84085819746075\) | \(-7384795740903483110476800\) | \([2]\) | \(2949120\) | \(3.4585\) |
Rank
sage: E.rank()
The elliptic curves in class 17640cb have rank \(0\).
Complex multiplication
The elliptic curves in class 17640cb do not have complex multiplication.Modular form 17640.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
