Show commands:
SageMath
E = EllipticCurve("ko1")
E.isogeny_class()
Elliptic curves in class 244800ko
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 244800.ko6 | 244800ko1 | \([0, 0, 0, -1152300, -584962000]\) | \(-56667352321/16711680\) | \(-49900809093120000000\) | \([2]\) | \(4718592\) | \(2.4942\) | \(\Gamma_0(N)\)-optimal |
| 244800.ko5 | 244800ko2 | \([0, 0, 0, -19584300, -33357058000]\) | \(278202094583041/16646400\) | \(49705884057600000000\) | \([2, 2]\) | \(9437184\) | \(2.8408\) | |
| 244800.ko4 | 244800ko3 | \([0, 0, 0, -20736300, -29212162000]\) | \(330240275458561/67652010000\) | \(202007819427840000000000\) | \([2, 2]\) | \(18874368\) | \(3.1873\) | |
| 244800.ko2 | 244800ko4 | \([0, 0, 0, -313344300, -2134916098000]\) | \(1139466686381936641/4080\) | \(12182814720000000\) | \([2]\) | \(18874368\) | \(3.1873\) | |
| 244800.ko3 | 244800ko5 | \([0, 0, 0, -103968300, 382120382000]\) | \(41623544884956481/2962701562500\) | \(8846579462400000000000000\) | \([2, 2]\) | \(37748736\) | \(3.5339\) | |
| 244800.ko7 | 244800ko6 | \([0, 0, 0, 44063700, -175271362000]\) | \(3168685387909439/6278181696900\) | \(-18746550096036249600000000\) | \([2]\) | \(37748736\) | \(3.5339\) | |
| 244800.ko1 | 244800ko7 | \([0, 0, 0, -1633968300, 25422100382000]\) | \(161572377633716256481/914742821250\) | \(2731407428367360000000000\) | \([2]\) | \(75497472\) | \(3.8805\) | |
| 244800.ko8 | 244800ko8 | \([0, 0, 0, 94319700, 1667423198000]\) | \(31077313442863199/420227050781250\) | \(-1254791250000000000000000000\) | \([2]\) | \(75497472\) | \(3.8805\) |
Rank
sage: E.rank()
The elliptic curves in class 244800ko have rank \(1\).
Complex multiplication
The elliptic curves in class 244800ko do not have complex multiplication.Modular form 244800.2.a.ko
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.
