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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 17850.ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17850.ba1 | 17850u5 | \([1, 0, 1, -43564501, -110677905352]\) | \(585196747116290735872321/836876053125000\) | \(13076188330078125000\) | \([2]\) | \(1769472\) | \(2.9394\) | |
| 17850.ba2 | 17850u4 | \([1, 0, 1, -6315501, 6107356648]\) | \(1782900110862842086081/328139630024640\) | \(5127181719135000000\) | \([2]\) | \(884736\) | \(2.5928\) | |
| 17850.ba3 | 17850u3 | \([1, 0, 1, -2747501, -1696515352]\) | \(146796951366228945601/5397929064360000\) | \(84342641630625000000\) | \([2, 2]\) | \(884736\) | \(2.5928\) | |
| 17850.ba4 | 17850u2 | \([1, 0, 1, -435501, 74476648]\) | \(584614687782041281/184812061593600\) | \(2887688462400000000\) | \([2, 2]\) | \(442368\) | \(2.2462\) | |
| 17850.ba5 | 17850u1 | \([1, 0, 1, 76499, 7916648]\) | \(3168685387909439/3563732336640\) | \(-55683317760000000\) | \([2]\) | \(221184\) | \(1.8997\) | \(\Gamma_0(N)\)-optimal |
| 17850.ba6 | 17850u6 | \([1, 0, 1, 1077499, -6049365352]\) | \(8854313460877886399/1016927675429790600\) | \(-15889494928590478125000\) | \([2]\) | \(1769472\) | \(2.9394\) |
Rank
sage: E.rank()
The elliptic curves in class 17850.ba have rank \(1\).
Complex multiplication
The elliptic curves in class 17850.ba do not have complex multiplication.Modular form 17850.2.a.ba
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
