Show commands:
SageMath
E = EllipticCurve("ik1")
E.isogeny_class()
Elliptic curves in class 177870ik
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177870.j5 | 177870ik1 | \([1, 1, 0, 29522, 7583668]\) | \(13651919/126720\) | \(-26411284004878080\) | \([2]\) | \(1474560\) | \(1.8323\) | \(\Gamma_0(N)\)-optimal |
| 177870.j4 | 177870ik2 | \([1, 1, 0, -444798, 105388452]\) | \(46694890801/3920400\) | \(817099098900915600\) | \([2, 2]\) | \(2949120\) | \(2.1788\) | |
| 177870.j2 | 177870ik3 | \([1, 1, 0, -6966698, 7074690792]\) | \(179415687049201/1443420\) | \(300841031868064380\) | \([2]\) | \(5898240\) | \(2.5254\) | |
| 177870.j3 | 177870ik4 | \([1, 1, 0, -1512018, -594921312]\) | \(1834216913521/329422500\) | \(68659021504868602500\) | \([2, 2]\) | \(5898240\) | \(2.5254\) | |
| 177870.j6 | 177870ik5 | \([1, 1, 0, 2934732, -3429279762]\) | \(13411719834479/32153832150\) | \(-6701578225685208061350\) | \([2]\) | \(11796480\) | \(2.8720\) | |
| 177870.j1 | 177870ik6 | \([1, 1, 0, -23034288, -42559043358]\) | \(6484907238722641/283593750\) | \(59107284353364843750\) | \([2]\) | \(11796480\) | \(2.8720\) |
Rank
sage: E.rank()
The elliptic curves in class 177870ik have rank \(1\).
Complex multiplication
The elliptic curves in class 177870ik do not have complex multiplication.Modular form 177870.2.a.ik
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.
