Show commands:
SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 53550u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 53550.n3 | 53550u1 | \([1, -1, 0, -4049442, -3136914284]\) | \(-644706081631626841/347004000000\) | \(-3952592437500000000\) | \([2]\) | \(1769472\) | \(2.5176\) | \(\Gamma_0(N)\)-optimal |
| 53550.n2 | 53550u2 | \([1, -1, 0, -64799442, -200756664284]\) | \(2641739317048851306841/764694000\) | \(8710342593750000\) | \([2]\) | \(3538944\) | \(2.8642\) | |
| 53550.n4 | 53550u3 | \([1, -1, 0, 3291183, -12746298659]\) | \(346124368852751159/6361262220902400\) | \(-72458752484966400000000\) | \([2]\) | \(5308416\) | \(3.0669\) | |
| 53550.n1 | 53550u4 | \([1, -1, 0, -65828817, -194048058659]\) | \(2769646315294225853641/174474906948464640\) | \(1987378236959855040000000\) | \([2]\) | \(10616832\) | \(3.4135\) |
Rank
sage: E.rank()
The elliptic curves in class 53550u have rank \(0\).
Complex multiplication
The elliptic curves in class 53550u do not have complex multiplication.Modular form 53550.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
