Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 69938e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69938.f4 | 69938e1 | \([1, 1, 0, -105635, -8225587]\) | \(3048625/1088\) | \(46524159352227392\) | \([2]\) | \(622080\) | \(1.8992\) | \(\Gamma_0(N)\)-optimal |
| 69938.f3 | 69938e2 | \([1, 1, 0, -1504395, -710682859]\) | \(8805624625/2312\) | \(98863838623483208\) | \([2]\) | \(1244160\) | \(2.2457\) | |
| 69938.f2 | 69938e3 | \([1, 1, 0, -3602535, 2629975649]\) | \(120920208625/19652\) | \(840342628299607268\) | \([2]\) | \(1866240\) | \(2.4485\) | |
| 69938.f1 | 69938e4 | \([1, 1, 0, -3952225, 2088165963]\) | \(159661140625/48275138\) | \(2064301666417985253842\) | \([2]\) | \(3732480\) | \(2.7950\) |
Rank
sage: E.rank()
The elliptic curves in class 69938e have rank \(0\).
Complex multiplication
The elliptic curves in class 69938e do not have complex multiplication.Modular form 69938.2.a.e
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.
