Show commands:
SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 98022q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 98022.n5 | 98022q1 | \([1, 1, 1, -32694, -2123853]\) | \(4354703137/352512\) | \(312855697596672\) | \([2]\) | \(460800\) | \(1.5244\) | \(\Gamma_0(N)\)-optimal |
| 98022.n4 | 98022q2 | \([1, 1, 1, -109574, 11468531]\) | \(163936758817/30338064\) | \(26925143474413584\) | \([2, 2]\) | \(921600\) | \(1.8709\) | |
| 98022.n6 | 98022q3 | \([1, 1, 1, 217166, 67145027]\) | \(1276229915423/2927177028\) | \(-2597880387288640068\) | \([2]\) | \(1843200\) | \(2.2175\) | |
| 98022.n2 | 98022q4 | \([1, 1, 1, -1666394, 827242211]\) | \(576615941610337/27060804\) | \(24016563160819524\) | \([2, 2]\) | \(1843200\) | \(2.2175\) | |
| 98022.n3 | 98022q5 | \([1, 1, 1, -1579904, 917053427]\) | \(-491411892194497/125563633938\) | \(-111438187319711525778\) | \([2]\) | \(3686400\) | \(2.5641\) | |
| 98022.n1 | 98022q6 | \([1, 1, 1, -26662004, 52978082915]\) | \(2361739090258884097/5202\) | \(4616794148562\) | \([2]\) | \(3686400\) | \(2.5641\) |
Rank
sage: E.rank()
The elliptic curves in class 98022q have rank \(1\).
Complex multiplication
The elliptic curves in class 98022q do not have complex multiplication.Modular form 98022.2.a.q
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
