Show commands:
SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 117600co
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 117600.gl3 | 117600co1 | \([0, 1, 0, -12658, -493312]\) | \(1906624/225\) | \(26471025000000\) | \([2, 2]\) | \(294912\) | \(1.3069\) | \(\Gamma_0(N)\)-optimal |
| 117600.gl4 | 117600co2 | \([0, 1, 0, 17967, -2483937]\) | \(85184/405\) | \(-3049462080000000\) | \([2]\) | \(589824\) | \(1.6535\) | |
| 117600.gl2 | 117600co3 | \([0, 1, 0, -49408, 3696188]\) | \(14172488/1875\) | \(1764735000000000\) | \([2]\) | \(589824\) | \(1.6535\) | |
| 117600.gl1 | 117600co4 | \([0, 1, 0, -196408, -33568312]\) | \(890277128/15\) | \(14117880000000\) | \([2]\) | \(589824\) | \(1.6535\) |
Rank
sage: E.rank()
The elliptic curves in class 117600co have rank \(1\).
Complex multiplication
The elliptic curves in class 117600co do not have complex multiplication.Modular form 117600.2.a.co
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
