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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 60690bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 60690.bv5 | 60690bw1 | \([1, 0, 0, 884334, 310979076]\) | \(3168685387909439/3563732336640\) | \(-86019835173179228160\) | \([2]\) | \(2654208\) | \(2.5115\) | \(\Gamma_0(N)\)-optimal |
| 60690.bv4 | 60690bw2 | \([1, 0, 0, -5034386, 2928237060]\) | \(584614687782041281/184812061593600\) | \(4460913888747769958400\) | \([2, 2]\) | \(5308416\) | \(2.8581\) | |
| 60690.bv3 | 60690bw3 | \([1, 0, 0, -31761106, -66673487164]\) | \(146796951366228945601/5397929064360000\) | \(130292885248094940840000\) | \([2, 2]\) | \(10616832\) | \(3.2047\) | |
| 60690.bv2 | 60690bw4 | \([1, 0, 0, -73007186, 240058147140]\) | \(1782900110862842086081/328139630024640\) | \(7920492961354219700160\) | \([2]\) | \(10616832\) | \(3.2047\) | |
| 60690.bv6 | 60690bw5 | \([1, 0, 0, 12455894, -237766746964]\) | \(8854313460877886399/1016927675429790600\) | \(-24546161933696175263051400\) | \([2]\) | \(21233664\) | \(3.5513\) | |
| 60690.bv1 | 60690bw6 | \([1, 0, 0, -503605626, -4349983670820]\) | \(585196747116290735872321/836876053125000\) | \(20200153476752353125000\) | \([2]\) | \(21233664\) | \(3.5513\) |
Rank
sage: E.rank()
The elliptic curves in class 60690bw have rank \(0\).
Complex multiplication
The elliptic curves in class 60690bw do not have complex multiplication.Modular form 60690.2.a.bw
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
