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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 56784bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 56784.u4 | 56784bw1 | \([0, -1, 0, 1127, 4588]\) | \(2048000/1323\) | \(-102173892912\) | \([2]\) | \(55296\) | \(0.80140\) | \(\Gamma_0(N)\)-optimal |
| 56784.u3 | 56784bw2 | \([0, -1, 0, -4788, 42444]\) | \(9826000/5103\) | \(6305588819712\) | \([2]\) | \(110592\) | \(1.1480\) | |
| 56784.u2 | 56784bw3 | \([0, -1, 0, -19153, 1057120]\) | \(-10061824000/352947\) | \(-27257724097968\) | \([2]\) | \(165888\) | \(1.3507\) | |
| 56784.u1 | 56784bw4 | \([0, -1, 0, -308988, 66212028]\) | \(2640279346000/3087\) | \(3814492002048\) | \([2]\) | \(331776\) | \(1.6973\) |
Rank
sage: E.rank()
The elliptic curves in class 56784bw have rank \(1\).
Complex multiplication
The elliptic curves in class 56784bw do not have complex multiplication.Modular form 56784.2.a.bw
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
