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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 66759.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 66759.e1 | 66759g6 | \([1, 0, 0, -1305997, -574570678]\) | \(10206027697760497/5557167\) | \(134136501907023\) | \([2]\) | \(819200\) | \(2.0394\) | |
| 66759.e2 | 66759g4 | \([1, 0, 0, -82082, -8877165]\) | \(2533811507137/58110129\) | \(1402637248336401\) | \([2, 2]\) | \(409600\) | \(1.6928\) | |
| 66759.e3 | 66759g2 | \([1, 0, 0, -11277, 256680]\) | \(6570725617/2614689\) | \(63112236151041\) | \([2, 2]\) | \(204800\) | \(1.3462\) | |
| 66759.e4 | 66759g1 | \([1, 0, 0, -9832, 374303]\) | \(4354703137/1617\) | \(39030449073\) | \([2]\) | \(102400\) | \(0.99968\) | \(\Gamma_0(N)\)-optimal |
| 66759.e5 | 66759g5 | \([1, 0, 0, 8953, -27466512]\) | \(3288008303/13504609503\) | \(-325968443696718207\) | \([2]\) | \(819200\) | \(2.0394\) | |
| 66759.e6 | 66759g3 | \([1, 0, 0, 36408, 1868433]\) | \(221115865823/190238433\) | \(-4591893302989377\) | \([2]\) | \(409600\) | \(1.6928\) |
Rank
sage: E.rank()
The elliptic curves in class 66759.e have rank \(1\).
Complex multiplication
The elliptic curves in class 66759.e do not have complex multiplication.Modular form 66759.2.a.e
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.
