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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 84.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84.b1 | 84a4 | \([0, 1, 0, -1828, -30700]\) | \(2640279346000/3087\) | \(790272\) | \([2]\) | \(36\) | \(0.41481\) | |
| 84.b2 | 84a3 | \([0, 1, 0, -113, -516]\) | \(-10061824000/352947\) | \(-5647152\) | \([2]\) | \(18\) | \(0.068235\) | |
| 84.b3 | 84a2 | \([0, 1, 0, -28, -28]\) | \(9826000/5103\) | \(1306368\) | \([6]\) | \(12\) | \(-0.13450\) | |
| 84.b4 | 84a1 | \([0, 1, 0, 7, 0]\) | \(2048000/1323\) | \(-21168\) | \([6]\) | \(6\) | \(-0.48107\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 84.b have rank \(0\).
Complex multiplication
The elliptic curves in class 84.b do not have complex multiplication.Modular form 84.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
