Show commands:
SageMath
E = EllipticCurve("bx1")
E.isogeny_class()
Elliptic curves in class 74970bx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
74970.bf6 | 74970bx1 | \([1, -1, 0, -35289, 3143853]\) | \(-56667352321/16711680\) | \(-1433295968993280\) | \([2]\) | \(393216\) | \(1.6227\) | \(\Gamma_0(N)\)-optimal |
74970.bf5 | 74970bx2 | \([1, -1, 0, -599769, 178922925]\) | \(278202094583041/16646400\) | \(1427697156614400\) | \([2, 2]\) | \(786432\) | \(1.9693\) | |
74970.bf4 | 74970bx3 | \([1, -1, 0, -635049, 156717693]\) | \(330240275458561/67652010000\) | \(5802250475553210000\) | \([2, 2]\) | \(1572864\) | \(2.3159\) | |
74970.bf2 | 74970bx4 | \([1, -1, 0, -9596169, 11444215005]\) | \(1139466686381936641/4080\) | \(349925773680\) | \([2]\) | \(1572864\) | \(2.3159\) | |
74970.bf7 | 74970bx5 | \([1, -1, 0, 1349451, 939007593]\) | \(3168685387909439/6278181696900\) | \(-538455291076310724900\) | \([2]\) | \(3145728\) | \(2.6624\) | |
74970.bf3 | 74970bx6 | \([1, -1, 0, -3184029, -2047130415]\) | \(41623544884956481/2962701562500\) | \(254099420696264062500\) | \([2, 2]\) | \(3145728\) | \(2.6624\) | |
74970.bf8 | 74970bx7 | \([1, -1, 0, 2888541, -8937068337]\) | \(31077313442863199/420227050781250\) | \(-36041244084777832031250\) | \([2]\) | \(6291456\) | \(3.0090\) | |
74970.bf1 | 74970bx8 | \([1, -1, 0, -50040279, -136234059165]\) | \(161572377633716256481/914742821250\) | \(78453943491208871250\) | \([2]\) | \(6291456\) | \(3.0090\) |
Rank
sage: E.rank()
The elliptic curves in class 74970bx have rank \(1\).
Complex multiplication
The elliptic curves in class 74970bx do not have complex multiplication.Modular form 74970.2.a.bx
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.