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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 11760cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 11760.bp7 | 11760cd1 | \([0, 1, 0, -390056, 93598644]\) | \(13619385906841/6048000\) | \(2914472558592000\) | \([2]\) | \(110592\) | \(1.9259\) | \(\Gamma_0(N)\)-optimal |
| 11760.bp6 | 11760cd2 | \([0, 1, 0, -452776, 61410740]\) | \(21302308926361/8930250000\) | \(4303400887296000000\) | \([2, 2]\) | \(221184\) | \(2.2725\) | |
| 11760.bp5 | 11760cd3 | \([0, 1, 0, -1154456, -363230316]\) | \(353108405631241/86318776320\) | \(41596181361752801280\) | \([2]\) | \(331776\) | \(2.4752\) | |
| 11760.bp4 | 11760cd4 | \([0, 1, 0, -3416296, -2388827596]\) | \(9150443179640281/184570312500\) | \(88942644000000000000\) | \([2]\) | \(442368\) | \(2.6191\) | |
| 11760.bp8 | 11760cd5 | \([0, 1, 0, 1507224, 452626740]\) | \(785793873833639/637994920500\) | \(-307443566190200832000\) | \([2]\) | \(442368\) | \(2.6191\) | |
| 11760.bp2 | 11760cd6 | \([0, 1, 0, -17210776, -27485566060]\) | \(1169975873419524361/108425318400\) | \(52249109645072793600\) | \([2, 2]\) | \(663552\) | \(2.8218\) | |
| 11760.bp1 | 11760cd7 | \([0, 1, 0, -275366296, -1758883007596]\) | \(4791901410190533590281/41160000\) | \(19834604912640000\) | \([2]\) | \(1327104\) | \(3.1684\) | |
| 11760.bp3 | 11760cd8 | \([0, 1, 0, -15956376, -31660711020]\) | \(-932348627918877961/358766164249920\) | \(-172885935955307880775680\) | \([2]\) | \(1327104\) | \(3.1684\) |
Rank
sage: E.rank()
The elliptic curves in class 11760cd have rank \(0\).
Complex multiplication
The elliptic curves in class 11760cd do not have complex multiplication.Modular form 11760.2.a.cd
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.
