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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 39984dk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 39984.cu2 | 39984dk1 | \([0, 1, 0, -200328, 34431732]\) | \(1845026709625/793152\) | \(382212258398208\) | \([2]\) | \(207360\) | \(1.7582\) | \(\Gamma_0(N)\)-optimal |
| 39984.cu3 | 39984dk2 | \([0, 1, 0, -168968, 45608436]\) | \(-1107111813625/1228691592\) | \(-592094564791123968\) | \([2]\) | \(414720\) | \(2.1048\) | |
| 39984.cu1 | 39984dk3 | \([0, 1, 0, -588408, -131539500]\) | \(46753267515625/11591221248\) | \(5585697130929979392\) | \([2]\) | \(622080\) | \(2.3075\) | |
| 39984.cu4 | 39984dk4 | \([0, 1, 0, 1418632, -832397868]\) | \(655215969476375/1001033261568\) | \(-482388222731115036672\) | \([2]\) | \(1244160\) | \(2.6541\) |
Rank
sage: E.rank()
The elliptic curves in class 39984dk have rank \(1\).
Complex multiplication
The elliptic curves in class 39984dk do not have complex multiplication.Modular form 39984.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
