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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 99099.bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 99099.bk1 | 99099q3 | \([0, 0, 1, -127776, -43328376]\) | \(-178643795968/524596891\) | \(-677500081363484379\) | \([]\) | \(1244160\) | \(2.1084\) | |
| 99099.bk2 | 99099q1 | \([0, 0, 1, -7986, 275184]\) | \(-43614208/91\) | \(-117523585179\) | \([]\) | \(138240\) | \(1.0097\) | \(\Gamma_0(N)\)-optimal |
| 99099.bk3 | 99099q2 | \([0, 0, 1, 13794, 1365273]\) | \(224755712/753571\) | \(-973212808867299\) | \([]\) | \(414720\) | \(1.5591\) |
Rank
sage: E.rank()
The elliptic curves in class 99099.bk have rank \(1\).
Complex multiplication
The elliptic curves in class 99099.bk do not have complex multiplication.Modular form 99099.2.a.bk
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
