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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 60690.bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 60690.bk1 | 60690bj8 | \([1, 1, 1, -555111206, -5034288025747]\) | \(783736670177727068275201/360150\) | \(8693145475350\) | \([2]\) | \(10485760\) | \(3.2110\) | |
| 60690.bk2 | 60690bj6 | \([1, 1, 1, -34694456, -78671565547]\) | \(191342053882402567201/129708022500\) | \(3130836342947302500\) | \([2, 2]\) | \(5242880\) | \(2.8645\) | |
| 60690.bk3 | 60690bj7 | \([1, 1, 1, -34477706, -79702775347]\) | \(-187778242790732059201/4984939585440150\) | \(-120324323204393015995350\) | \([2]\) | \(10485760\) | \(3.2110\) | |
| 60690.bk4 | 60690bj4 | \([1, 1, 1, -4355236, 3495591989]\) | \(378499465220294881/120530818800\) | \(2909320955411497200\) | \([2]\) | \(2621440\) | \(2.5179\) | |
| 60690.bk5 | 60690bj3 | \([1, 1, 1, -2181956, -1213785547]\) | \(47595748626367201/1215506250000\) | \(29339365979306250000\) | \([2, 2]\) | \(2621440\) | \(2.5179\) | |
| 60690.bk6 | 60690bj2 | \([1, 1, 1, -309236, 38689589]\) | \(135487869158881/51438240000\) | \(1241594067238560000\) | \([2, 2]\) | \(1310720\) | \(2.1713\) | |
| 60690.bk7 | 60690bj1 | \([1, 1, 1, 60684, 4361013]\) | \(1023887723039/928972800\) | \(-22423145059123200\) | \([2]\) | \(655360\) | \(1.8247\) | \(\Gamma_0(N)\)-optimal |
| 60690.bk8 | 60690bj5 | \([1, 1, 1, 367024, -3876959851]\) | \(226523624554079/269165039062500\) | \(-6496989702758789062500\) | \([2]\) | \(5242880\) | \(2.8645\) |
Rank
sage: E.rank()
The elliptic curves in class 60690.bk have rank \(1\).
Complex multiplication
The elliptic curves in class 60690.bk do not have complex multiplication.Modular form 60690.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
