Show commands:
SageMath
E = EllipticCurve("ff1")
E.isogeny_class()
Elliptic curves in class 61200ff
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 61200.ek6 | 61200ff1 | \([0, 0, 0, -288075, 73120250]\) | \(-56667352321/16711680\) | \(-779700142080000000\) | \([2]\) | \(589824\) | \(2.1476\) | \(\Gamma_0(N)\)-optimal |
| 61200.ek5 | 61200ff2 | \([0, 0, 0, -4896075, 4169632250]\) | \(278202094583041/16646400\) | \(776654438400000000\) | \([2, 2]\) | \(1179648\) | \(2.4942\) | |
| 61200.ek4 | 61200ff3 | \([0, 0, 0, -5184075, 3651520250]\) | \(330240275458561/67652010000\) | \(3156372178560000000000\) | \([2, 2]\) | \(2359296\) | \(2.8408\) | |
| 61200.ek2 | 61200ff4 | \([0, 0, 0, -78336075, 266864512250]\) | \(1139466686381936641/4080\) | \(190356480000000\) | \([2]\) | \(2359296\) | \(2.8408\) | |
| 61200.ek7 | 61200ff5 | \([0, 0, 0, 11015925, 21908920250]\) | \(3168685387909439/6278181696900\) | \(-292914845250566400000000\) | \([2]\) | \(4718592\) | \(3.1873\) | |
| 61200.ek3 | 61200ff6 | \([0, 0, 0, -25992075, -47765047750]\) | \(41623544884956481/2962701562500\) | \(138227804100000000000000\) | \([2, 2]\) | \(4718592\) | \(3.1873\) | |
| 61200.ek8 | 61200ff7 | \([0, 0, 0, 23579925, -208427899750]\) | \(31077313442863199/420227050781250\) | \(-19606113281250000000000000\) | \([2]\) | \(9437184\) | \(3.5339\) | |
| 61200.ek1 | 61200ff8 | \([0, 0, 0, -408492075, -3177762547750]\) | \(161572377633716256481/914742821250\) | \(42678241068240000000000\) | \([2]\) | \(9437184\) | \(3.5339\) |
Rank
sage: E.rank()
The elliptic curves in class 61200ff have rank \(0\).
Complex multiplication
The elliptic curves in class 61200ff do not have complex multiplication.Modular form 61200.2.a.ff
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
