Show commands:
SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 8400cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8400.ce7 | 8400cc1 | \([0, 1, 0, 83992, -7032012]\) | \(1023887723039/928972800\) | \(-59454259200000000\) | \([2]\) | \(73728\) | \(1.9060\) | \(\Gamma_0(N)\)-optimal |
| 8400.ce6 | 8400cc2 | \([0, 1, 0, -428008, -63352012]\) | \(135487869158881/51438240000\) | \(3292047360000000000\) | \([2, 2]\) | \(147456\) | \(2.2526\) | |
| 8400.ce4 | 8400cc3 | \([0, 1, 0, -6028008, -5696952012]\) | \(378499465220294881/120530818800\) | \(7713972403200000000\) | \([2]\) | \(294912\) | \(2.5991\) | |
| 8400.ce5 | 8400cc4 | \([0, 1, 0, -3020008, 1973959988]\) | \(47595748626367201/1215506250000\) | \(77792400000000000000\) | \([2, 2]\) | \(294912\) | \(2.5991\) | |
| 8400.ce2 | 8400cc5 | \([0, 1, 0, -48020008, 128063959988]\) | \(191342053882402567201/129708022500\) | \(8301313440000000000\) | \([2, 2]\) | \(589824\) | \(2.9457\) | |
| 8400.ce8 | 8400cc6 | \([0, 1, 0, 507992, 6313399988]\) | \(226523624554079/269165039062500\) | \(-17226562500000000000000\) | \([2]\) | \(589824\) | \(2.9457\) | |
| 8400.ce1 | 8400cc7 | \([0, 1, 0, -768320008, 8196864559988]\) | \(783736670177727068275201/360150\) | \(23049600000000\) | \([2]\) | \(1179648\) | \(3.2923\) | |
| 8400.ce3 | 8400cc8 | \([0, 1, 0, -47720008, 129743359988]\) | \(-187778242790732059201/4984939585440150\) | \(-319036133468169600000000\) | \([2]\) | \(1179648\) | \(3.2923\) |
Rank
sage: E.rank()
The elliptic curves in class 8400cc have rank \(1\).
Complex multiplication
The elliptic curves in class 8400cc do not have complex multiplication.Modular form 8400.2.a.cc
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
