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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 4002.n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4002.n1 | 4002o3 | \([1, 0, 0, -242219, -45904101]\) | \(1571623248760107387697/12708699174\) | \(12708699174\) | \([2]\) | \(23040\) | \(1.5281\) | |
| 4002.n2 | 4002o4 | \([1, 0, 0, -21359, -75057]\) | \(1077625178826324337/621306044897562\) | \(621306044897562\) | \([2]\) | \(23040\) | \(1.5281\) | |
| 4002.n3 | 4002o2 | \([1, 0, 0, -15149, -717171]\) | \(384483228869610577/1091026208484\) | \(1091026208484\) | \([2, 2]\) | \(11520\) | \(1.1815\) | |
| 4002.n4 | 4002o1 | \([1, 0, 0, -569, -20247]\) | \(-20375497153297/164474612208\) | \(-164474612208\) | \([4]\) | \(5760\) | \(0.83497\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4002.n have rank \(0\).
Complex multiplication
The elliptic curves in class 4002.n do not have complex multiplication.Modular form 4002.2.a.n
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.
