Show commands:
SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 76050dk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
76050.fo3 | 76050dk1 | \([1, -1, 1, -32480, 2574147]\) | \(-1860867/320\) | \(-651619215000000\) | \([2]\) | \(414720\) | \(1.5688\) | \(\Gamma_0(N)\)-optimal |
76050.fo2 | 76050dk2 | \([1, -1, 1, -539480, 152646147]\) | \(8527173507/200\) | \(407262009375000\) | \([2]\) | \(829440\) | \(1.9154\) | |
76050.fo4 | 76050dk3 | \([1, -1, 1, 221020, -10945853]\) | \(804357/500\) | \(-742235012085937500\) | \([2]\) | \(1244160\) | \(2.1181\) | |
76050.fo1 | 76050dk4 | \([1, -1, 1, -919730, -88516853]\) | \(57960603/31250\) | \(46389688255371093750\) | \([2]\) | \(2488320\) | \(2.4647\) |
Rank
sage: E.rank()
The elliptic curves in class 76050dk have rank \(0\).
Complex multiplication
The elliptic curves in class 76050dk do not have complex multiplication.Modular form 76050.2.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.