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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 6006h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6006.d4 | 6006h1 | \([1, 1, 0, 1859, 467005]\) | \(709899390552743/94315065901056\) | \(-94315065901056\) | \([2]\) | \(21504\) | \(1.3608\) | \(\Gamma_0(N)\)-optimal |
| 6006.d3 | 6006h2 | \([1, 1, 0, -80061, 8413245]\) | \(56753835752958457177/2022564252893184\) | \(2022564252893184\) | \([2, 2]\) | \(43008\) | \(1.7074\) | |
| 6006.d2 | 6006h3 | \([1, 1, 0, -201021, -23205699]\) | \(898362003697422318937/288927801457206912\) | \(288927801457206912\) | \([2]\) | \(86016\) | \(2.0540\) | |
| 6006.d1 | 6006h4 | \([1, 1, 0, -1269821, 550229949]\) | \(226439278116330906299737/303624334645632\) | \(303624334645632\) | \([2]\) | \(86016\) | \(2.0540\) |
Rank
sage: E.rank()
The elliptic curves in class 6006h have rank \(1\).
Complex multiplication
The elliptic curves in class 6006h do not have complex multiplication.Modular form 6006.2.a.h
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.
