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SageMath
sage: E = EllipticCurve("e1")
sage: E.isogeny_class()
Elliptic curves in class 55440.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 55440.e1 | 55440ct6 | [0, 0, 0, -439084803, -3541365261502] | [2] | 3932160 | |
| 55440.e2 | 55440ct4 | [0, 0, 0, -27442803, -55333820302] | [2, 2] | 1966080 | |
| 55440.e3 | 55440ct5 | [0, 0, 0, -27306723, -55909738078] | [2] | 3932160 | |
| 55440.e4 | 55440ct3 | [0, 0, 0, -3664083, 1423613522] | [2] | 1966080 | |
| 55440.e5 | 55440ct2 | [0, 0, 0, -1723683, -855580318] | [2, 2] | 983040 | |
| 55440.e6 | 55440ct1 | [0, 0, 0, 5037, -39970222] | [2] | 491520 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55440.e have rank \(1\).
Complex multiplication
The elliptic curves in class 55440.e do not have complex multiplication.Modular form 55440.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
