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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 59850.bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 59850.bt1 | 59850m3 | \([1, -1, 0, -31430742, -67815619084]\) | \(11165451838341046875/572244736\) | \(175992080292000000\) | \([2]\) | \(3981312\) | \(2.7821\) | |
| 59850.bt2 | 59850m4 | \([1, -1, 0, -31376742, -68060293084]\) | \(-11108001800138902875/79947274872976\) | \(-24587534551949790750000\) | \([2]\) | \(7962624\) | \(3.1286\) | |
| 59850.bt3 | 59850m1 | \([1, -1, 0, -422742, -75299084]\) | \(19804628171203875/5638671302656\) | \(2378814455808000000\) | \([2]\) | \(1327104\) | \(2.2328\) | \(\Gamma_0(N)\)-optimal |
| 59850.bt4 | 59850m2 | \([1, -1, 0, 1113258, -497699084]\) | \(361682234074684125/462672528510976\) | \(-195189972965568000000\) | \([2]\) | \(2654208\) | \(2.5793\) |
Rank
sage: E.rank()
The elliptic curves in class 59850.bt have rank \(0\).
Complex multiplication
The elliptic curves in class 59850.bt do not have complex multiplication.Modular form 59850.2.a.bt
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
