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SageMath
E = EllipticCurve("en1")
E.isogeny_class()
Elliptic curves in class 33600.en
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33600.en1 | 33600cd8 | \([0, 1, 0, -561972033, -5127858515937]\) | \(4791901410190533590281/41160000\) | \(168591360000000000\) | \([2]\) | \(5308416\) | \(3.3467\) | |
| 33600.en2 | 33600cd6 | \([0, 1, 0, -35124033, -80127827937]\) | \(1169975873419524361/108425318400\) | \(444110104166400000000\) | \([2, 2]\) | \(2654208\) | \(3.0001\) | |
| 33600.en3 | 33600cd7 | \([0, 1, 0, -32564033, -92300627937]\) | \(-932348627918877961/358766164249920\) | \(-1469506208767672320000000\) | \([2]\) | \(5308416\) | \(3.3467\) | |
| 33600.en4 | 33600cd5 | \([0, 1, 0, -6972033, -6963515937]\) | \(9150443179640281/184570312500\) | \(756000000000000000000\) | \([2]\) | \(1769472\) | \(2.7974\) | |
| 33600.en5 | 33600cd3 | \([0, 1, 0, -2356033, -1058643937]\) | \(353108405631241/86318776320\) | \(353561707806720000000\) | \([2]\) | \(1327104\) | \(2.6536\) | |
| 33600.en6 | 33600cd2 | \([0, 1, 0, -924033, 179172063]\) | \(21302308926361/8930250000\) | \(36578304000000000000\) | \([2, 2]\) | \(884736\) | \(2.4508\) | |
| 33600.en7 | 33600cd1 | \([0, 1, 0, -796033, 272996063]\) | \(13619385906841/6048000\) | \(24772608000000000\) | \([2]\) | \(442368\) | \(2.1043\) | \(\Gamma_0(N)\)-optimal |
| 33600.en8 | 33600cd4 | \([0, 1, 0, 3075967, 1319172063]\) | \(785793873833639/637994920500\) | \(-2613227194368000000000\) | \([2]\) | \(1769472\) | \(2.7974\) |
Rank
sage: E.rank()
The elliptic curves in class 33600.en have rank \(0\).
Complex multiplication
The elliptic curves in class 33600.en do not have complex multiplication.Modular form 33600.2.a.en
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
