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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 58800bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 58800.eo5 | 58800bd1 | \([0, -1, 0, -18783, 995562]\) | \(24918016/45\) | \(1323551250000\) | \([2]\) | \(147456\) | \(1.2180\) | \(\Gamma_0(N)\)-optimal |
| 58800.eo4 | 58800bd2 | \([0, -1, 0, -24908, 297312]\) | \(3631696/2025\) | \(952956900000000\) | \([2, 2]\) | \(294912\) | \(1.5645\) | |
| 58800.eo6 | 58800bd3 | \([0, -1, 0, 97592, 2257312]\) | \(54607676/32805\) | \(-61751607120000000\) | \([2]\) | \(589824\) | \(1.9111\) | |
| 58800.eo2 | 58800bd4 | \([0, -1, 0, -245408, -46448688]\) | \(868327204/5625\) | \(10588410000000000\) | \([2, 2]\) | \(589824\) | \(1.9111\) | |
| 58800.eo3 | 58800bd5 | \([0, -1, 0, -98408, -101720688]\) | \(-27995042/1171875\) | \(-4411837500000000000\) | \([2]\) | \(1179648\) | \(2.2577\) | |
| 58800.eo1 | 58800bd6 | \([0, -1, 0, -3920408, -2986448688]\) | \(1770025017602/75\) | \(282357600000000\) | \([2]\) | \(1179648\) | \(2.2577\) |
Rank
sage: E.rank()
The elliptic curves in class 58800bd have rank \(0\).
Complex multiplication
The elliptic curves in class 58800bd do not have complex multiplication.Modular form 58800.2.a.bd
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
