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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 66990bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 66990.bg7 | 66990bf1 | \([1, 0, 1, -7604139, -7854138938]\) | \(48626585041073592629153449/1495685481508052326800\) | \(1495685481508052326800\) | \([6]\) | \(4866048\) | \(2.8388\) | \(\Gamma_0(N)\)-optimal |
| 66990.bg6 | 66990bf2 | \([1, 0, 1, -18232959, 18943242046]\) | \(670340013887004175017226729/232881726498686518522500\) | \(232881726498686518522500\) | \([2, 6]\) | \(9732096\) | \(3.1853\) | |
| 66990.bg3 | 66990bf3 | \([1, 0, 1, -611314554, -5817669148244]\) | \(25264919424633968163234466954009/788559976512000000\) | \(788559976512000000\) | \([2]\) | \(14598144\) | \(3.3881\) | |
| 66990.bg8 | 66990bf4 | \([1, 0, 1, 54280671, 132151521202]\) | \(17687210581982180689098502391/17779596413363063064843750\) | \(-17779596413363063064843750\) | \([6]\) | \(19464192\) | \(3.5319\) | |
| 66990.bg5 | 66990bf5 | \([1, 0, 1, -260807709, 1620809861146]\) | \(1961936660078092398490361110729/474945809169127845190950\) | \(474945809169127845190950\) | \([6]\) | \(19464192\) | \(3.5319\) | |
| 66990.bg2 | 66990bf6 | \([1, 0, 1, -611340474, -5817151142228]\) | \(25268133291250118646200025886489/4463489141983265625000000\) | \(4463489141983265625000000\) | \([2, 2]\) | \(29196288\) | \(3.7346\) | |
| 66990.bg4 | 66990bf7 | \([1, 0, 1, -549880194, -7033032153524]\) | \(-18387722572391758931230103324569/10726928227901458740234375000\) | \(-10726928227901458740234375000\) | \([2]\) | \(58392576\) | \(4.0812\) | |
| 66990.bg1 | 66990bf8 | \([1, 0, 1, -673215474, -4568117642228]\) | \(33743177813840232088099335886489/10512677519144598197082375000\) | \(10512677519144598197082375000\) | \([2]\) | \(58392576\) | \(4.0812\) |
Rank
sage: E.rank()
The elliptic curves in class 66990bf have rank \(0\).
Complex multiplication
The elliptic curves in class 66990bf do not have complex multiplication.Modular form 66990.2.a.bf
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
