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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 81225bg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 81225.f4 | 81225bg1 | \([1, -1, 1, 1825870, -4293199128]\) | \(1256216039/15582375\) | \(-8350314096900568359375\) | \([2]\) | \(4976640\) | \(2.8872\) | \(\Gamma_0(N)\)-optimal |
| 81225.f3 | 81225bg2 | \([1, -1, 1, -31070255, -62256171378]\) | \(6189976379881/456890625\) | \(244839456544924072265625\) | \([2, 2]\) | \(9953280\) | \(3.2338\) | |
| 81225.f2 | 81225bg3 | \([1, -1, 1, -100517630, 314148601122]\) | \(209595169258201/41748046875\) | \(22372026365581512451171875\) | \([2]\) | \(19906560\) | \(3.5804\) | |
| 81225.f1 | 81225bg4 | \([1, -1, 1, -487960880, -4148685921378]\) | \(23977812996389881/146611125\) | \(78566261166860080078125\) | \([2]\) | \(19906560\) | \(3.5804\) |
Rank
sage: E.rank()
The elliptic curves in class 81225bg have rank \(1\).
Complex multiplication
The elliptic curves in class 81225bg do not have complex multiplication.Modular form 81225.2.a.bg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
