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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 4851.p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4851.p1 | 4851o5 | \([1, -1, 0, -1992888, -1082361771]\) | \(10206027697760497/5557167\) | \(476616657339207\) | \([2]\) | \(61440\) | \(2.1451\) | |
| 4851.p2 | 4851o3 | \([1, -1, 0, -125253, -16689240]\) | \(2533811507137/58110129\) | \(4983880355139609\) | \([2, 2]\) | \(30720\) | \(1.7985\) | |
| 4851.p3 | 4851o2 | \([1, -1, 0, -17208, 489915]\) | \(6570725617/2614689\) | \(224251733151369\) | \([2, 2]\) | \(15360\) | \(1.4519\) | |
| 4851.p4 | 4851o1 | \([1, -1, 0, -15003, 710856]\) | \(4354703137/1617\) | \(138683817657\) | \([2]\) | \(7680\) | \(1.1053\) | \(\Gamma_0(N)\)-optimal |
| 4851.p5 | 4851o6 | \([1, -1, 0, 13662, -51779169]\) | \(3288008303/13504609503\) | \(-1158237972692047863\) | \([2]\) | \(61440\) | \(2.1451\) | |
| 4851.p6 | 4851o4 | \([1, -1, 0, 55557, 3502386]\) | \(221115865823/190238433\) | \(-16316012463528393\) | \([2]\) | \(30720\) | \(1.7985\) |
Rank
sage: E.rank()
The elliptic curves in class 4851.p have rank \(0\).
Complex multiplication
The elliptic curves in class 4851.p do not have complex multiplication.Modular form 4851.2.a.p
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
