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SageMath
E = EllipticCurve("gn1")
E.isogeny_class()
Elliptic curves in class 235200gn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 235200.gn7 | 235200gn1 | \([0, -1, 0, -39005633, -93715660863]\) | \(13619385906841/6048000\) | \(2914472558592000000000\) | \([2]\) | \(21233664\) | \(3.0772\) | \(\Gamma_0(N)\)-optimal |
| 235200.gn6 | 235200gn2 | \([0, -1, 0, -45277633, -61546572863]\) | \(21302308926361/8930250000\) | \(4303400887296000000000000\) | \([2, 2]\) | \(42467328\) | \(3.4238\) | |
| 235200.gn5 | 235200gn3 | \([0, -1, 0, -115445633, 362883979137]\) | \(353108405631241/86318776320\) | \(41596181361752801280000000\) | \([2]\) | \(63700992\) | \(3.6265\) | |
| 235200.gn4 | 235200gn4 | \([0, -1, 0, -341629633, 2387802707137]\) | \(9150443179640281/184570312500\) | \(88942644000000000000000000\) | \([2]\) | \(84934656\) | \(3.7704\) | |
| 235200.gn8 | 235200gn5 | \([0, -1, 0, 150722367, -452174572863]\) | \(785793873833639/637994920500\) | \(-307443566190200832000000000\) | \([2]\) | \(84934656\) | \(3.7704\) | |
| 235200.gn2 | 235200gn6 | \([0, -1, 0, -1721077633, 27480402827137]\) | \(1169975873419524361/108425318400\) | \(52249109645072793600000000\) | \([2, 2]\) | \(127401984\) | \(3.9731\) | |
| 235200.gn1 | 235200gn7 | \([0, -1, 0, -27536629633, 1758800397707137]\) | \(4791901410190533590281/41160000\) | \(19834604912640000000000\) | \([2]\) | \(254803968\) | \(4.3197\) | |
| 235200.gn3 | 235200gn8 | \([0, -1, 0, -1595637633, 31655924107137]\) | \(-932348627918877961/358766164249920\) | \(-172885935955307880775680000000\) | \([2]\) | \(254803968\) | \(4.3197\) |
Rank
sage: E.rank()
The elliptic curves in class 235200gn have rank \(0\).
Complex multiplication
The elliptic curves in class 235200gn do not have complex multiplication.Modular form 235200.2.a.gn
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.
