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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 4410.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4410.bf1 | 4410z4 | \([1, -1, 1, -10667, 113509]\) | \(57960603/31250\) | \(72365164593750\) | \([2]\) | \(17280\) | \(1.3505\) | |
| 4410.bf2 | 4410z2 | \([1, -1, 1, -6257, -188919]\) | \(8527173507/200\) | \(635304600\) | \([2]\) | \(5760\) | \(0.80117\) | |
| 4410.bf3 | 4410z1 | \([1, -1, 1, -377, -3111]\) | \(-1860867/320\) | \(-1016487360\) | \([2]\) | \(2880\) | \(0.45459\) | \(\Gamma_0(N)\)-optimal |
| 4410.bf4 | 4410z3 | \([1, -1, 1, 2563, 12961]\) | \(804357/500\) | \(-1157842633500\) | \([2]\) | \(8640\) | \(1.0039\) |
Rank
sage: E.rank()
The elliptic curves in class 4410.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 4410.bf do not have complex multiplication.Modular form 4410.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
