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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 17787o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17787.w4 | 17787o1 | \([1, 0, 1, -201710, 34840859]\) | \(4354703137/1617\) | \(337018988603913\) | \([2]\) | \(115200\) | \(1.7550\) | \(\Gamma_0(N)\)-optimal |
| 17787.w3 | 17787o2 | \([1, 0, 1, -231355, 23919641]\) | \(6570725617/2614689\) | \(544959704572527321\) | \([2, 2]\) | \(230400\) | \(2.1015\) | |
| 17787.w2 | 17787o3 | \([1, 0, 1, -1683960, -824401679]\) | \(2533811507137/58110129\) | \(12111451393458821481\) | \([2, 2]\) | \(460800\) | \(2.4481\) | |
| 17787.w6 | 17787o4 | \([1, 0, 1, 746930, 173401589]\) | \(221115865823/190238433\) | \(-39649946990261760537\) | \([2]\) | \(460800\) | \(2.4481\) | |
| 17787.w1 | 17787o5 | \([1, 0, 1, -26793275, -53383219837]\) | \(10206027697760497/5557167\) | \(1158237972692047863\) | \([2]\) | \(921600\) | \(2.7947\) | |
| 17787.w5 | 17787o6 | \([1, 0, 1, 183675, -2552337581]\) | \(3288008303/13504609503\) | \(-2814662854787787385767\) | \([2]\) | \(921600\) | \(2.7947\) |
Rank
sage: E.rank()
The elliptic curves in class 17787o have rank \(0\).
Complex multiplication
The elliptic curves in class 17787o do not have complex multiplication.Modular form 17787.2.a.o
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.
