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SageMath
E = EllipticCurve("le1")
E.isogeny_class()
Elliptic curves in class 342720le
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 342720.le5 | 342720le1 | \([0, 0, 0, 20148, -2724496]\) | \(4733169839/19518975\) | \(-3730134210969600\) | \([2]\) | \(2097152\) | \(1.6706\) | \(\Gamma_0(N)\)-optimal |
| 342720.le4 | 342720le2 | \([0, 0, 0, -213132, -33424144]\) | \(5602762882081/716900625\) | \(137001842933760000\) | \([2, 2]\) | \(4194304\) | \(2.0171\) | |
| 342720.le3 | 342720le3 | \([0, 0, 0, -861132, 273209456]\) | \(369543396484081/45120132225\) | \(8622591545711001600\) | \([2, 2]\) | \(8388608\) | \(2.3637\) | |
| 342720.le2 | 342720le4 | \([0, 0, 0, -3297612, -2304835216]\) | \(20751759537944401/418359375\) | \(79949721600000000\) | \([2]\) | \(8388608\) | \(2.3637\) | |
| 342720.le1 | 342720le5 | \([0, 0, 0, -13345932, 18765695216]\) | \(1375634265228629281/24990412335\) | \(4775742168685608960\) | \([2]\) | \(16777216\) | \(2.7103\) | |
| 342720.le6 | 342720le6 | \([0, 0, 0, 1255668, 1405274096]\) | \(1145725929069119/5127181719135\) | \(-979819685019494645760\) | \([2]\) | \(16777216\) | \(2.7103\) |
Rank
sage: E.rank()
The elliptic curves in class 342720le have rank \(1\).
Complex multiplication
The elliptic curves in class 342720le do not have complex multiplication.Modular form 342720.2.a.le
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
