Show commands:
SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 82467.be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 82467.be1 | 82467bd4 | \([1, -1, 0, -159841341, -552408900408]\) | \(5265932508006615127873/1510137598013239041\) | \(129518643957852819192329961\) | \([2]\) | \(17694720\) | \(3.7170\) | |
| 82467.be2 | 82467bd2 | \([1, -1, 0, -59643936, 170515376667]\) | \(273594167224805799793/11903648120953281\) | \(1020929725083101733593001\) | \([2, 2]\) | \(8847360\) | \(3.3704\) | |
| 82467.be3 | 82467bd1 | \([1, -1, 0, -59006691, 174476109240]\) | \(264918160154242157473/536027170833\) | \(45972971192950748793\) | \([2]\) | \(4423680\) | \(3.0238\) | \(\Gamma_0(N)\)-optimal |
| 82467.be4 | 82467bd3 | \([1, -1, 0, 30357549, 639945122130]\) | \(36075142039228937567/2083708275110728497\) | \(-178711576051848028671850137\) | \([2]\) | \(17694720\) | \(3.7170\) |
Rank
sage: E.rank()
The elliptic curves in class 82467.be have rank \(1\).
Complex multiplication
The elliptic curves in class 82467.be do not have complex multiplication.Modular form 82467.2.a.be
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.
