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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 25992bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25992.w5 | 25992bc1 | \([0, 0, 0, 2166, 48013]\) | \(2048/3\) | \(-1646229467952\) | \([2]\) | \(27648\) | \(1.0296\) | \(\Gamma_0(N)\)-optimal |
| 25992.w4 | 25992bc2 | \([0, 0, 0, -14079, 480130]\) | \(35152/9\) | \(79019014461696\) | \([2, 2]\) | \(55296\) | \(1.3762\) | |
| 25992.w3 | 25992bc3 | \([0, 0, 0, -79059, -8162210]\) | \(1556068/81\) | \(2844684520621056\) | \([2, 2]\) | \(110592\) | \(1.7227\) | |
| 25992.w2 | 25992bc4 | \([0, 0, 0, -209019, 36777958]\) | \(28756228/3\) | \(105358685948928\) | \([2]\) | \(110592\) | \(1.7227\) | |
| 25992.w6 | 25992bc5 | \([0, 0, 0, 50901, -32360762]\) | \(207646/6561\) | \(-460838892340611072\) | \([2]\) | \(221184\) | \(2.0693\) | |
| 25992.w1 | 25992bc6 | \([0, 0, 0, -1248699, -537073418]\) | \(3065617154/9\) | \(632152115693568\) | \([2]\) | \(221184\) | \(2.0693\) |
Rank
sage: E.rank()
The elliptic curves in class 25992bc have rank \(0\).
Complex multiplication
The elliptic curves in class 25992bc do not have complex multiplication.Modular form 25992.2.a.bc
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.
