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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1040.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1040.e1 | 1040f3 | \([0, 0, 0, -22187, 1272026]\) | \(294889639316481/260\) | \(1064960\) | \([4]\) | \(768\) | \(0.88893\) | |
1040.e2 | 1040f2 | \([0, 0, 0, -1387, 19866]\) | \(72043225281/67600\) | \(276889600\) | \([2, 2]\) | \(384\) | \(0.54236\) | |
1040.e3 | 1040f4 | \([0, 0, 0, -1067, 29274]\) | \(-32798729601/71402500\) | \(-292464640000\) | \([4]\) | \(768\) | \(0.88893\) | |
1040.e4 | 1040f1 | \([0, 0, 0, -107, 154]\) | \(33076161/16640\) | \(68157440\) | \([2]\) | \(192\) | \(0.19579\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1040.e have rank \(0\).
Complex multiplication
The elliptic curves in class 1040.e do not have complex multiplication.Modular form 1040.2.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.