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SageMath
E = EllipticCurve("bs1")
E.isogeny_class()
Elliptic curves in class 476850bs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 476850.bs5 | 476850bs1 | \([1, 1, 0, -2052050, 856168500]\) | \(2533811507137/625016832\) | \(235724795446272000000\) | \([2]\) | \(18874368\) | \(2.6195\) | \(\Gamma_0(N)\)-optimal |
| 476850.bs4 | 476850bs2 | \([1, 1, 0, -11300050, -13912887500]\) | \(423108074414017/23284318464\) | \(8781669430355844000000\) | \([2, 2]\) | \(37748736\) | \(2.9661\) | |
| 476850.bs6 | 476850bs3 | \([1, 1, 0, 7773950, -56085501500]\) | \(137763859017023/3683199928848\) | \(-1389117069115057664250000\) | \([2]\) | \(75497472\) | \(3.3127\) | |
| 476850.bs2 | 476850bs4 | \([1, 1, 0, -178342050, -916774897500]\) | \(1663303207415737537/5483698704\) | \(2068174310047040250000\) | \([2, 2]\) | \(75497472\) | \(3.3127\) | |
| 476850.bs3 | 476850bs5 | \([1, 1, 0, -175885550, -943253511000]\) | \(-1595514095015181697/95635786040388\) | \(-36068990381548470887062500\) | \([2]\) | \(150994944\) | \(3.6593\) | |
| 476850.bs1 | 476850bs6 | \([1, 1, 0, -2853470550, -58670124084000]\) | \(6812873765474836663297/74052\) | \(27928675931062500\) | \([2]\) | \(150994944\) | \(3.6593\) |
Rank
sage: E.rank()
The elliptic curves in class 476850bs have rank \(0\).
Complex multiplication
The elliptic curves in class 476850bs do not have complex multiplication.Modular form 476850.2.a.bs
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
