Show commands:
SageMath
E = EllipticCurve("ip1")
E.isogeny_class()
Elliptic curves in class 244800ip
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 244800.ip5 | 244800ip1 | \([0, 0, 0, -489900, -122402000]\) | \(4354703137/352512\) | \(1052595191808000000\) | \([2]\) | \(3145728\) | \(2.2011\) | \(\Gamma_0(N)\)-optimal |
| 244800.ip4 | 244800ip2 | \([0, 0, 0, -1641900, 667870000]\) | \(163936758817/30338064\) | \(90588973694976000000\) | \([2, 2]\) | \(6291456\) | \(2.5477\) | |
| 244800.ip2 | 244800ip3 | \([0, 0, 0, -24969900, 48023710000]\) | \(576615941610337/27060804\) | \(80803127771136000000\) | \([2, 2]\) | \(12582912\) | \(2.8942\) | |
| 244800.ip6 | 244800ip4 | \([0, 0, 0, 3254100, 3889438000]\) | \(1276229915423/2927177028\) | \(-8740503770775552000000\) | \([2]\) | \(12582912\) | \(2.8942\) | |
| 244800.ip1 | 244800ip5 | \([0, 0, 0, -399513900, 3073590142000]\) | \(2361739090258884097/5202\) | \(15533088768000000\) | \([2]\) | \(25165824\) | \(3.2408\) | |
| 244800.ip3 | 244800ip6 | \([0, 0, 0, -23673900, 53231038000]\) | \(-491411892194497/125563633938\) | \(-374931001920724992000000\) | \([2]\) | \(25165824\) | \(3.2408\) |
Rank
sage: E.rank()
The elliptic curves in class 244800ip have rank \(1\).
Complex multiplication
The elliptic curves in class 244800ip do not have complex multiplication.Modular form 244800.2.a.ip
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
