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SageMath
E = EllipticCurve("bx1")
E.isogeny_class()
Elliptic curves in class 301665.bx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 301665.bx1 | 301665bx6 | \([1, 0, 1, -3915734, -2982692659]\) | \(1375634265228629281/24990412335\) | \(120623947172289015\) | \([2]\) | \(7864320\) | \(2.4037\) | |
| 301665.bx2 | 301665bx3 | \([1, 0, 1, -967529, 366218777]\) | \(20751759537944401/418359375\) | \(2019340796484375\) | \([2]\) | \(3932160\) | \(2.0572\) | |
| 301665.bx3 | 301665bx4 | \([1, 0, 1, -252659, -43441279]\) | \(369543396484081/45120132225\) | \(217786260304820025\) | \([2, 2]\) | \(3932160\) | \(2.0572\) | |
| 301665.bx4 | 301665bx2 | \([1, 0, 1, -62534, 5306771]\) | \(5602762882081/716900625\) | \(3460342388855625\) | \([2, 2]\) | \(1966080\) | \(1.7106\) | |
| 301665.bx5 | 301665bx1 | \([1, 0, 1, 5911, 433487]\) | \(4733169839/19518975\) | \(-94214364200775\) | \([2]\) | \(983040\) | \(1.3640\) | \(\Gamma_0(N)\)-optimal |
| 301665.bx6 | 301665bx5 | \([1, 0, 1, 368416, -223304599]\) | \(1145725929069119/5127181719135\) | \(-24747926866556290215\) | \([2]\) | \(7864320\) | \(2.4037\) |
Rank
sage: E.rank()
The elliptic curves in class 301665.bx have rank \(0\).
Complex multiplication
The elliptic curves in class 301665.bx do not have complex multiplication.Modular form 301665.2.a.bx
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
