Show commands:
SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 4080bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4080.x3 | 4080bb1 | \([0, 1, 0, -1616, -22380]\) | \(114013572049/15667200\) | \(64172851200\) | \([2]\) | \(4608\) | \(0.80053\) | \(\Gamma_0(N)\)-optimal |
| 4080.x2 | 4080bb2 | \([0, 1, 0, -6736, 188564]\) | \(8253429989329/936360000\) | \(3835330560000\) | \([2, 2]\) | \(9216\) | \(1.1471\) | |
| 4080.x1 | 4080bb3 | \([0, 1, 0, -104656, 12996500]\) | \(30949975477232209/478125000\) | \(1958400000000\) | \([2]\) | \(18432\) | \(1.4937\) | |
| 4080.x4 | 4080bb4 | \([0, 1, 0, 9264, 962964]\) | \(21464092074671/109596256200\) | \(-448906265395200\) | \([4]\) | \(18432\) | \(1.4937\) |
Rank
sage: E.rank()
The elliptic curves in class 4080bb have rank \(0\).
Complex multiplication
The elliptic curves in class 4080bb do not have complex multiplication.Modular form 4080.2.a.bb
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.
