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SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 100800ed
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 100800.w4 | 100800ed1 | \([0, 0, 0, -661575, 862373500]\) | \(-43927191786304/415283203125\) | \(-302741455078125000000\) | \([2]\) | \(2949120\) | \(2.6123\) | \(\Gamma_0(N)\)-optimal |
| 100800.w3 | 100800ed2 | \([0, 0, 0, -18239700, 29901436000]\) | \(14383655824793536/45209390625\) | \(2109289329000000000000\) | \([2, 2]\) | \(5898240\) | \(2.9589\) | |
| 100800.w2 | 100800ed3 | \([0, 0, 0, -26114700, 1567186000]\) | \(5276930158229192/3050936350875\) | \(1138755891091392000000000\) | \([2]\) | \(11796480\) | \(3.3055\) | |
| 100800.w1 | 100800ed4 | \([0, 0, 0, -291614700, 1916735686000]\) | \(7347751505995469192/72930375\) | \(27221116608000000000\) | \([2]\) | \(11796480\) | \(3.3055\) |
Rank
sage: E.rank()
The elliptic curves in class 100800ed have rank \(0\).
Complex multiplication
The elliptic curves in class 100800ed do not have complex multiplication.Modular form 100800.2.a.ed
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 Cremona 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 Cremona labels.
