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SageMath
E = EllipticCurve("cb1")
E.isogeny_class()
Elliptic curves in class 333270cb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 333270.cb7 | 333270cb1 | \([1, -1, 0, 999711, 289659645]\) | \(1023887723039/928972800\) | \(-100253038128213196800\) | \([2]\) | \(11534336\) | \(2.5252\) | \(\Gamma_0(N)\)-optimal |
| 333270.cb6 | 333270cb2 | \([1, -1, 0, -5094369, 2596878333]\) | \(135487869158881/51438240000\) | \(5551120372919617440000\) | \([2, 2]\) | \(23068672\) | \(2.8718\) | |
| 333270.cb4 | 333270cb3 | \([1, -1, 0, -71748369, 233872927533]\) | \(378499465220294881/120530818800\) | \(13007464559544860722800\) | \([2]\) | \(46137344\) | \(3.2183\) | |
| 333270.cb5 | 333270cb4 | \([1, -1, 0, -35945649, -81090303795]\) | \(47595748626367201/1215506250000\) | \(131175201713474756250000\) | \([2, 2]\) | \(46137344\) | \(3.2183\) | |
| 333270.cb8 | 333270cb5 | \([1, -1, 0, 6046371, -259245647847]\) | \(226523624554079/269165039062500\) | \(-29047796581250610351562500\) | \([2]\) | \(92274688\) | \(3.5649\) | |
| 333270.cb2 | 333270cb6 | \([1, -1, 0, -571558149, -5259284831295]\) | \(191342053882402567201/129708022500\) | \(13997851525069017322500\) | \([2, 2]\) | \(92274688\) | \(3.5649\) | |
| 333270.cb3 | 333270cb7 | \([1, -1, 0, -567987399, -5328243869445]\) | \(-187778242790732059201/4984939585440150\) | \(-537965523130462640865102150\) | \([2]\) | \(184549376\) | \(3.9115\) | |
| 333270.cb1 | 333270cb8 | \([1, -1, 0, -9144928899, -336601202903145]\) | \(783736670177727068275201/360150\) | \(38866726433622150\) | \([2]\) | \(184549376\) | \(3.9115\) |
Rank
sage: E.rank()
The elliptic curves in class 333270cb have rank \(1\).
Complex multiplication
The elliptic curves in class 333270cb do not have complex multiplication.Modular form 333270.2.a.cb
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
