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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 116160.bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 116160.bk1 | 116160r6 | \([0, -1, 0, -1548961, 742524961]\) | \(1770025017602/75\) | \(17415153254400\) | \([2]\) | \(1310720\) | \(2.0255\) | |
| 116160.bk2 | 116160r4 | \([0, -1, 0, -96961, 11588161]\) | \(868327204/5625\) | \(653068247040000\) | \([2, 2]\) | \(655360\) | \(1.6789\) | |
| 116160.bk3 | 116160r5 | \([0, -1, 0, -38881, 25283425]\) | \(-27995042/1171875\) | \(-272111769600000000\) | \([2]\) | \(1310720\) | \(2.0255\) | |
| 116160.bk4 | 116160r2 | \([0, -1, 0, -9841, -68495]\) | \(3631696/2025\) | \(58776142233600\) | \([2, 2]\) | \(327680\) | \(1.3324\) | |
| 116160.bk5 | 116160r1 | \([0, -1, 0, -7421, -243219]\) | \(24918016/45\) | \(81633530880\) | \([2]\) | \(163840\) | \(0.98580\) | \(\Gamma_0(N)\)-optimal |
| 116160.bk6 | 116160r3 | \([0, -1, 0, 38559, -581535]\) | \(54607676/32805\) | \(-3808694016737280\) | \([2]\) | \(655360\) | \(1.6789\) |
Rank
sage: E.rank()
The elliptic curves in class 116160.bk have rank \(0\).
Complex multiplication
The elliptic curves in class 116160.bk do not have complex multiplication.Modular form 116160.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
