Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 283920d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 283920.d7 | 283920d1 | \([0, -1, 0, 567784, -124275984]\) | \(1023887723039/928972800\) | \(-18366358617273139200\) | \([2]\) | \(7077888\) | \(2.3838\) | \(\Gamma_0(N)\)-optimal |
| 283920.d6 | 283920d2 | \([0, -1, 0, -2893336, -1110002960]\) | \(135487869158881/51438240000\) | \(1016965364843151360000\) | \([2, 2]\) | \(14155776\) | \(2.7303\) | |
| 283920.d5 | 283920d3 | \([0, -1, 0, -20415256, 34718819056]\) | \(47595748626367201/1215506250000\) | \(24031299612902400000000\) | \([2, 2]\) | \(28311552\) | \(3.0769\) | |
| 283920.d4 | 283920d4 | \([0, -1, 0, -40749336, -100080729360]\) | \(378499465220294881/120530818800\) | \(2382967770977112883200\) | \([2]\) | \(28311552\) | \(3.0769\) | |
| 283920.d2 | 283920d5 | \([0, -1, 0, -324615256, 2251241699056]\) | \(191342053882402567201/129708022500\) | \(2564406683136829440000\) | \([2, 2]\) | \(56623104\) | \(3.4235\) | |
| 283920.d8 | 283920d6 | \([0, -1, 0, 3434024, 110960197360]\) | \(226523624554079/269165039062500\) | \(-5321556922500000000000000\) | \([2]\) | \(56623104\) | \(3.4235\) | |
| 283920.d1 | 283920d7 | \([0, -1, 0, -5193843256, 144074324118256]\) | \(783736670177727068275201/360150\) | \(7120385070489600\) | \([2]\) | \(113246208\) | \(3.7701\) | |
| 283920.d3 | 283920d8 | \([0, -1, 0, -322587256, 2280756399856]\) | \(-187778242790732059201/4984939585440150\) | \(-98555294742359183283609600\) | \([2]\) | \(113246208\) | \(3.7701\) |
Rank
sage: E.rank()
The elliptic curves in class 283920d have rank \(0\).
Complex multiplication
The elliptic curves in class 283920d do not have complex multiplication.Modular form 283920.2.a.d
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
