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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 8034.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8034.j1 | 8034j3 | \([1, 0, 0, -202397, -35064003]\) | \(916929740159167290193/4266578796948\) | \(4266578796948\) | \([2]\) | \(46080\) | \(1.6263\) | |
| 8034.j2 | 8034j4 | \([1, 0, 0, -39637, 2389205]\) | \(6886946780408974033/1512692597392236\) | \(1512692597392236\) | \([2]\) | \(46080\) | \(1.6263\) | |
| 8034.j3 | 8034j2 | \([1, 0, 0, -12857, -529815]\) | \(235041610922990353/15245307666576\) | \(15245307666576\) | \([2, 2]\) | \(23040\) | \(1.2797\) | |
| 8034.j4 | 8034j1 | \([1, 0, 0, 663, -34983]\) | \(32227258038767/549007310592\) | \(-549007310592\) | \([4]\) | \(11520\) | \(0.93313\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8034.j have rank \(0\).
Complex multiplication
The elliptic curves in class 8034.j do not have complex multiplication.Modular form 8034.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
