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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 52371.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 52371.a1 | 52371e4 | \([1, -1, 1, -697586, 224210310]\) | \(347873904937/395307\) | \(42660805293060867\) | \([2]\) | \(540672\) | \(2.1044\) | |
| 52371.a2 | 52371e2 | \([1, -1, 1, -54851, 1566906]\) | \(169112377/88209\) | \(9519353247211929\) | \([2, 2]\) | \(270336\) | \(1.7578\) | |
| 52371.a3 | 52371e1 | \([1, -1, 1, -31046, -2080020]\) | \(30664297/297\) | \(32051694435057\) | \([2]\) | \(135168\) | \(1.4113\) | \(\Gamma_0(N)\)-optimal |
| 52371.a4 | 52371e3 | \([1, -1, 1, 207004, 12041106]\) | \(9090072503/5845851\) | \(-630873501565226931\) | \([2]\) | \(540672\) | \(2.1044\) |
Rank
sage: E.rank()
The elliptic curves in class 52371.a have rank \(2\).
Complex multiplication
The elliptic curves in class 52371.a do not have complex multiplication.Modular form 52371.2.a.a
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.
