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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 133584bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 133584.j6 | 133584bn1 | \([0, -1, 0, 59976, -16595280]\) | \(3288008303/18259263\) | \(-132494943107248128\) | \([2]\) | \(983040\) | \(1.9675\) | \(\Gamma_0(N)\)-optimal |
| 133584.j5 | 133584bn2 | \([0, -1, 0, -724104, -213556176]\) | \(5786435182177/627352209\) | \(4552264526758907904\) | \([2, 2]\) | \(1966080\) | \(2.3141\) | |
| 133584.j4 | 133584bn3 | \([0, -1, 0, -2727864, 1504868400]\) | \(309368403125137/44372288367\) | \(321979250899889713152\) | \([2]\) | \(3932160\) | \(2.6606\) | |
| 133584.j2 | 133584bn4 | \([0, -1, 0, -11265624, -14550023376]\) | \(21790813729717297/304746849\) | \(2211338782971039744\) | \([2, 2]\) | \(3932160\) | \(2.6606\) | |
| 133584.j3 | 133584bn5 | \([0, -1, 0, -10946184, -15414300240]\) | \(-19989223566735457/2584262514273\) | \(-18752219889860567666688\) | \([2]\) | \(7864320\) | \(3.0072\) | |
| 133584.j1 | 133584bn6 | \([0, -1, 0, -180249384, -931388311632]\) | \(89254274298475942657/17457\) | \(126673470984192\) | \([2]\) | \(7864320\) | \(3.0072\) |
Rank
sage: E.rank()
The elliptic curves in class 133584bn have rank \(0\).
Complex multiplication
The elliptic curves in class 133584bn do not have complex multiplication.Modular form 133584.2.a.bn
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.
