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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 206910.do
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 206910.do1 | 206910bs4 | \([1, -1, 1, -57524731628, -5310419615972413]\) | \(16300610738133468173382620881/2228489100\) | \(2878022291915637900\) | \([2]\) | \(268800000\) | \(4.3571\) | |
| 206910.do2 | 206910bs3 | \([1, -1, 1, -3595295408, -82974647844349]\) | \(-3979640234041473454886161/1471455901872240\) | \(-1900338165064005090280560\) | \([2]\) | \(134400000\) | \(4.0105\) | |
| 206910.do3 | 206910bs2 | \([1, -1, 1, -95772128, -310782457213]\) | \(75224183150104868881/11219310000000000\) | \(14489379499281390000000000\) | \([2]\) | \(53760000\) | \(3.5524\) | |
| 206910.do4 | 206910bs1 | \([1, -1, 1, 10165792, -26529830269]\) | \(89962967236397039/287450726400000\) | \(-371233405811382681600000\) | \([2]\) | \(26880000\) | \(3.2058\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 206910.do have rank \(1\).
Complex multiplication
The elliptic curves in class 206910.do do not have complex multiplication.Modular form 206910.2.a.do
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
