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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 17986.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17986.e1 | 17986j4 | \([1, 0, 0, -59788, 3883374]\) | \(159661140625/48275138\) | \(7146452970427682\) | \([2]\) | \(152064\) | \(1.7472\) | |
| 17986.e2 | 17986j3 | \([1, 0, 0, -54498, 4891648]\) | \(120920208625/19652\) | \(2909201290628\) | \([2]\) | \(76032\) | \(1.4007\) | |
| 17986.e3 | 17986j2 | \([1, 0, 0, -22758, -1323044]\) | \(8805624625/2312\) | \(342258975368\) | \([2]\) | \(50688\) | \(1.1979\) | |
| 17986.e4 | 17986j1 | \([1, 0, 0, -1598, -15356]\) | \(3048625/1088\) | \(161063047232\) | \([2]\) | \(25344\) | \(0.85135\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 17986.e have rank \(0\).
Complex multiplication
The elliptic curves in class 17986.e do not have complex multiplication.Modular form 17986.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
