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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 199920bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 199920.eb5 | 199920bf1 | \([0, 1, 0, 27424, 4335540]\) | \(4733169839/19518975\) | \(-9406004796518400\) | \([2]\) | \(1572864\) | \(1.7476\) | \(\Gamma_0(N)\)-optimal |
| 199920.eb4 | 199920bf2 | \([0, 1, 0, -290096, 52979604]\) | \(5602762882081/716900625\) | \(345467460119040000\) | \([2, 2]\) | \(3145728\) | \(2.0942\) | |
| 199920.eb2 | 199920bf3 | \([0, 1, 0, -4488416, 3658496820]\) | \(20751759537944401/418359375\) | \(201603326400000000\) | \([2]\) | \(6291456\) | \(2.4408\) | |
| 199920.eb3 | 199920bf4 | \([0, 1, 0, -1172096, -434237196]\) | \(369543396484081/45120132225\) | \(21742954234425446400\) | \([2, 2]\) | \(6291456\) | \(2.4408\) | |
| 199920.eb6 | 199920bf5 | \([0, 1, 0, 1709104, -2230953516]\) | \(1145725929069119/5127181719135\) | \(-2470739157297207767040\) | \([2]\) | \(12582912\) | \(2.7874\) | |
| 199920.eb1 | 199920bf6 | \([0, 1, 0, -18165296, -29805284076]\) | \(1375634265228629281/24990412335\) | \(12042637397198499840\) | \([2]\) | \(12582912\) | \(2.7874\) |
Rank
sage: E.rank()
The elliptic curves in class 199920bf have rank \(2\).
Complex multiplication
The elliptic curves in class 199920bf do not have complex multiplication.Modular form 199920.2.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 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.
