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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1638e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1638.c4 | 1638e1 | \([1, -1, 0, 7797, -181819]\) | \(71903073502287/60782804992\) | \(-44310664839168\) | \([2]\) | \(5760\) | \(1.3062\) | \(\Gamma_0(N)\)-optimal |
| 1638.c3 | 1638e2 | \([1, -1, 0, -38283, -1573435]\) | \(8511781274893233/3440817243136\) | \(2508355770246144\) | \([2, 2]\) | \(11520\) | \(1.6527\) | |
| 1638.c1 | 1638e3 | \([1, -1, 0, -532203, -149255515]\) | \(22868021811807457713/8953460393696\) | \(6527072627004384\) | \([2]\) | \(23040\) | \(1.9993\) | |
| 1638.c2 | 1638e4 | \([1, -1, 0, -281643, 56492261]\) | \(3389174547561866673/74853681183008\) | \(54568333582412832\) | \([2]\) | \(23040\) | \(1.9993\) |
Rank
sage: E.rank()
The elliptic curves in class 1638e have rank \(0\).
Complex multiplication
The elliptic curves in class 1638e do not have complex multiplication.Modular form 1638.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
