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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 203280ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 203280.dg7 | 203280ck1 | \([0, -1, 0, -963200, 364032000]\) | \(13619385906841/6048000\) | \(43886186201088000\) | \([2]\) | \(3317760\) | \(2.1519\) | \(\Gamma_0(N)\)-optimal |
| 203280.dg6 | 203280ck2 | \([0, -1, 0, -1118080, 239260672]\) | \(21302308926361/8930250000\) | \(64800696812544000000\) | \([2, 2]\) | \(6635520\) | \(2.4985\) | |
| 203280.dg5 | 203280ck3 | \([0, -1, 0, -2850800, -1407059520]\) | \(353108405631241/86318776320\) | \(626356132643780689920\) | \([2]\) | \(9953280\) | \(2.7012\) | |
| 203280.dg8 | 203280ck4 | \([0, -1, 0, 3721920, 1753212672]\) | \(785793873833639/637994920500\) | \(-4629491381681768448000\) | \([2]\) | \(13271040\) | \(2.8451\) | |
| 203280.dg4 | 203280ck5 | \([0, -1, 0, -8436160, -9262534400]\) | \(9150443179640281/184570312500\) | \(1339300116000000000000\) | \([2]\) | \(13271040\) | \(2.8451\) | |
| 203280.dg2 | 203280ck6 | \([0, -1, 0, -42500080, -106620388928]\) | \(1169975873419524361/108425318400\) | \(786768140247131750400\) | \([2, 2]\) | \(19906560\) | \(3.0478\) | |
| 203280.dg3 | 203280ck7 | \([0, -1, 0, -39402480, -122824554048]\) | \(-932348627918877961/358766164249920\) | \(-2603319888710666342891520\) | \([2]\) | \(39813120\) | \(3.3944\) | |
| 203280.dg1 | 203280ck8 | \([0, -1, 0, -679986160, -6824703694400]\) | \(4791901410190533590281/41160000\) | \(298669878312960000\) | \([2]\) | \(39813120\) | \(3.3944\) |
Rank
sage: E.rank()
The elliptic curves in class 203280ck have rank \(1\).
Complex multiplication
The elliptic curves in class 203280ck do not have complex multiplication.Modular form 203280.2.a.ck
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.
