Show commands:
SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 5202.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5202.c1 | 5202b5 | \([1, -1, 0, -72162198, 235964108794]\) | \(2361739090258884097/5202\) | \(91535889140802\) | \([2]\) | \(294912\) | \(2.8130\) | |
| 5202.c2 | 5202b3 | \([1, -1, 0, -4510188, 3687697660]\) | \(576615941610337/27060804\) | \(476169695310452004\) | \([2, 2]\) | \(147456\) | \(2.4664\) | |
| 5202.c3 | 5202b6 | \([1, -1, 0, -4276098, 4087382926]\) | \(-491411892194497/125563633938\) | \(-2209453840112458990338\) | \([2]\) | \(294912\) | \(2.8130\) | |
| 5202.c4 | 5202b2 | \([1, -1, 0, -296568, 51343600]\) | \(163936758817/30338064\) | \(533837305469157264\) | \([2, 2]\) | \(73728\) | \(2.1198\) | |
| 5202.c5 | 5202b1 | \([1, -1, 0, -88488, -9374144]\) | \(4354703137/352512\) | \(6202902605306112\) | \([2]\) | \(36864\) | \(1.7733\) | \(\Gamma_0(N)\)-optimal |
| 5202.c6 | 5202b4 | \([1, -1, 0, 587772, 298428196]\) | \(1276229915423/2927177028\) | \(-51507449429163835428\) | \([2]\) | \(147456\) | \(2.4664\) |
Rank
sage: E.rank()
The elliptic curves in class 5202.c have rank \(0\).
Complex multiplication
The elliptic curves in class 5202.c do not have complex multiplication.Modular form 5202.2.a.c
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 LMFDB 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 LMFDB labels.
