Show commands:
SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 194208u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 194208.i3 | 194208u1 | \([0, -1, 0, -23794, -1233056]\) | \(964430272/127449\) | \(196883778014784\) | \([2, 2]\) | \(516096\) | \(1.4707\) | \(\Gamma_0(N)\)-optimal |
| 194208.i2 | 194208u2 | \([0, -1, 0, -97489, 10484449]\) | \(1036433728/122451\) | \(12106422114791424\) | \([4]\) | \(1032192\) | \(1.8173\) | |
| 194208.i4 | 194208u3 | \([0, -1, 0, 36896, -6549500]\) | \(449455096/1753941\) | \(-21675966417627648\) | \([2]\) | \(1032192\) | \(1.8173\) | |
| 194208.i1 | 194208u4 | \([0, -1, 0, -367704, -85697352]\) | \(444893916104/9639\) | \(119122958126592\) | \([2]\) | \(1032192\) | \(1.8173\) |
Rank
sage: E.rank()
The elliptic curves in class 194208u have rank \(1\).
Complex multiplication
The elliptic curves in class 194208u do not have complex multiplication.Modular form 194208.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
