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SageMath
E = EllipticCurve("fe1")
E.isogeny_class()
Elliptic curves in class 208080.fe
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 208080.fe1 | 208080o8 | \([0, 0, 0, -4722168387, -124898779176766]\) | \(161572377633716256481/914742821250\) | \(65929535269329709347840000\) | \([2]\) | \(113246208\) | \(4.1458\) | |
| 208080.fe2 | 208080o3 | \([0, 0, 0, -905565027, 10488842789474]\) | \(1139466686381936641/4080\) | \(294063530918215680\) | \([2]\) | \(28311552\) | \(3.4527\) | |
| 208080.fe3 | 208080o6 | \([0, 0, 0, -300468387, -1877357436766]\) | \(41623544884956481/2962701562500\) | \(213534922187662905600000000\) | \([2, 2]\) | \(56623104\) | \(3.7992\) | |
| 208080.fe4 | 208080o4 | \([0, 0, 0, -59927907, 143519351906]\) | \(330240275458561/67652010000\) | \(4875977679979028520960000\) | \([2, 2]\) | \(28311552\) | \(3.4527\) | |
| 208080.fe5 | 208080o2 | \([0, 0, 0, -56598627, 163883225954]\) | \(278202094583041/16646400\) | \(1199779206146319974400\) | \([2, 2]\) | \(14155776\) | \(3.1061\) | |
| 208080.fe6 | 208080o1 | \([0, 0, 0, -3330147, 2873918306]\) | \(-56667352321/16711680\) | \(-1204484222641011425280\) | \([2]\) | \(7077888\) | \(2.7595\) | \(\Gamma_0(N)\)-optimal |
| 208080.fe7 | 208080o5 | \([0, 0, 0, 127344093, 861108201506]\) | \(3168685387909439/6278181696900\) | \(-452496146455031601221222400\) | \([2]\) | \(56623104\) | \(3.7992\) | |
| 208080.fe8 | 208080o7 | \([0, 0, 0, 272583933, -8192050171774]\) | \(31077313442863199/420227050781250\) | \(-30287610377471250000000000000\) | \([2]\) | \(113246208\) | \(4.1458\) |
Rank
sage: E.rank()
The elliptic curves in class 208080.fe have rank \(0\).
Complex multiplication
The elliptic curves in class 208080.fe do not have complex multiplication.Modular form 208080.2.a.fe
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 8 & 4 \\ 16 & 1 & 8 & 4 & 2 & 4 & 8 & 16 \\ 2 & 8 & 1 & 2 & 4 & 8 & 4 & 2 \\ 4 & 4 & 2 & 1 & 2 & 4 & 2 & 4 \\ 8 & 2 & 4 & 2 & 1 & 2 & 4 & 8 \\ 16 & 4 & 8 & 4 & 2 & 1 & 8 & 16 \\ 8 & 8 & 4 & 2 & 4 & 8 & 1 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
