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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 277530bo
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 277530.bo6 | 277530bo1 | \([1, 0, 1, 214437, 3650278]\) | \(1833318007919/1070530560\) | \(-636776542931189760\) | \([2]\) | \(4300800\) | \(2.1057\) | \(\Gamma_0(N)\)-optimal |
| 277530.bo5 | 277530bo2 | \([1, 0, 1, -862043, 29055206]\) | \(119102750067601/68309049600\) | \(40631815737425721600\) | \([2, 2]\) | \(8601600\) | \(2.4523\) | |
| 277530.bo2 | 277530bo3 | \([1, 0, 1, -9944843, 12043783046]\) | \(182864522286982801/463015182960\) | \(275412228801689810160\) | \([2]\) | \(17203200\) | \(2.7988\) | |
| 277530.bo3 | 277530bo4 | \([1, 0, 1, -9002923, -10355451322]\) | \(135670761487282321/643043610000\) | \(382497335648028810000\) | \([2, 2]\) | \(17203200\) | \(2.7988\) | |
| 277530.bo4 | 277530bo5 | \([1, 0, 1, -4377423, -20981149922]\) | \(-15595206456730321/310672490129100\) | \(-184795242321930980741100\) | \([2]\) | \(34406400\) | \(3.1454\) | |
| 277530.bo1 | 277530bo6 | \([1, 0, 1, -143882503, -664305606994]\) | \(553808571467029327441/12529687500\) | \(7452950329842187500\) | \([2]\) | \(34406400\) | \(3.1454\) |
Rank
sage: E.rank()
The elliptic curves in class 277530bo have rank \(1\).
Complex multiplication
The elliptic curves in class 277530bo do not have complex multiplication.Modular form 277530.2.a.bo
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.
