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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1785.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1785.b1 | 1785h3 | \([1, 1, 1, -1015, 9182]\) | \(115650783909361/27072079335\) | \(27072079335\) | \([4]\) | \(1536\) | \(0.71365\) | |
| 1785.b2 | 1785h2 | \([1, 1, 1, -340, -2428]\) | \(4347507044161/258084225\) | \(258084225\) | \([2, 2]\) | \(768\) | \(0.36707\) | |
| 1785.b3 | 1785h1 | \([1, 1, 1, -335, -2500]\) | \(4158523459441/16065\) | \(16065\) | \([2]\) | \(384\) | \(0.020501\) | \(\Gamma_0(N)\)-optimal |
| 1785.b4 | 1785h4 | \([1, 1, 1, 255, -9330]\) | \(1833318007919/39525924375\) | \(-39525924375\) | \([2]\) | \(1536\) | \(0.71365\) |
Rank
sage: E.rank()
The elliptic curves in class 1785.b have rank \(0\).
Complex multiplication
The elliptic curves in class 1785.b do not have complex multiplication.Modular form 1785.2.a.b
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.
