Show commands:
SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 98736.dk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 98736.dk1 | 98736dj4 | \([0, 1, 0, -936906912, 11037508285428]\) | \(12534210458299016895673/315581882565708\) | \(2289961187164111790850048\) | \([4]\) | \(44236800\) | \(3.7824\) | |
| 98736.dk2 | 98736dj2 | \([0, 1, 0, -60789472, 158582809460]\) | \(3423676911662954233/483711578981136\) | \(3509963032868455519420416\) | \([2, 2]\) | \(22118400\) | \(3.4358\) | |
| 98736.dk3 | 98736dj1 | \([0, 1, 0, -16029152, -22230979212]\) | \(62768149033310713/6915442583808\) | \(50180621841258431643648\) | \([2]\) | \(11059200\) | \(3.0892\) | \(\Gamma_0(N)\)-optimal |
| 98736.dk4 | 98736dj3 | \([0, 1, 0, 99162848, 852455973620]\) | \(14861225463775641287/51859390496937804\) | \(-376308013826644512735412224\) | \([2]\) | \(44236800\) | \(3.7824\) |
Rank
sage: E.rank()
The elliptic curves in class 98736.dk have rank \(1\).
Complex multiplication
The elliptic curves in class 98736.dk do not have complex multiplication.Modular form 98736.2.a.dk
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.
