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SageMath
sage: E = EllipticCurve("d1")
sage: E.isogeny_class()
Elliptic curves in class 41070.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 41070.d1 | 41070b4 | [1, 1, 0, -466835873, 3882153327477] | [2] | 9455616 | |
| 41070.d2 | 41070b6 | [1, 1, 0, -414978153, -3239790411267] | [2] | 18911232 | |
| 41070.d3 | 41070b3 | [1, 1, 0, -40145953, 10979326453] | [2, 2] | 9455616 | |
| 41070.d4 | 41070b2 | [1, 1, 0, -29193953, 60576553653] | [2, 2] | 4727808 | |
| 41070.d5 | 41070b1 | [1, 1, 0, -1156833, 1648134837] | [2] | 2363904 | \(\Gamma_0(N)\)-optimal |
| 41070.d6 | 41070b5 | [1, 1, 0, 159454247, 87745563373] | [2] | 18911232 |
Rank
sage: E.rank()
The elliptic curves in class 41070.d have rank \(1\).
Complex multiplication
The elliptic curves in class 41070.d do not have complex multiplication.Modular form 41070.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
