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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 2736.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2736.e1 | 2736j3 | \([0, 0, 0, -11451, 458570]\) | \(111223479026/3518667\) | \(5253341681664\) | \([4]\) | \(3072\) | \(1.2155\) | |
| 2736.e2 | 2736j2 | \([0, 0, 0, -1731, -17710]\) | \(768400132/263169\) | \(196454605824\) | \([2, 2]\) | \(1536\) | \(0.86893\) | |
| 2736.e3 | 2736j1 | \([0, 0, 0, -1551, -23506]\) | \(2211014608/513\) | \(95738112\) | \([2]\) | \(768\) | \(0.52236\) | \(\Gamma_0(N)\)-optimal |
| 2736.e4 | 2736j4 | \([0, 0, 0, 5109, -123046]\) | \(9878111854/10097379\) | \(-15075306067968\) | \([2]\) | \(3072\) | \(1.2155\) |
Rank
sage: E.rank()
The elliptic curves in class 2736.e have rank \(1\).
Complex multiplication
The elliptic curves in class 2736.e do not have complex multiplication.Modular form 2736.2.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 LMFDB 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 LMFDB labels.
