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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 122199.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122199.f1 | 122199f6 | \([1, 1, 1, -2390562, 1421653488]\) | \(10206027697760497/5557167\) | \(822660157166463\) | \([2]\) | \(2027520\) | \(2.1905\) | |
122199.f2 | 122199f4 | \([1, 1, 1, -150247, 21904676]\) | \(2533811507137/58110129\) | \(8602384606419681\) | \([2, 2]\) | \(1013760\) | \(1.8440\) | |
122199.f3 | 122199f2 | \([1, 1, 1, -20642, -646594]\) | \(6570725617/2614689\) | \(387067810573521\) | \([2, 2]\) | \(506880\) | \(1.4974\) | |
122199.f4 | 122199f1 | \([1, 1, 1, -17997, -936486]\) | \(4354703137/1617\) | \(239374032513\) | \([2]\) | \(253440\) | \(1.1508\) | \(\Gamma_0(N)\)-optimal |
122199.f5 | 122199f5 | \([1, 1, 1, 16388, 68029244]\) | \(3288008303/13504609503\) | \(-1999166873374453167\) | \([2]\) | \(2027520\) | \(2.1905\) | |
122199.f6 | 122199f3 | \([1, 1, 1, 66643, -4591876]\) | \(221115865823/190238433\) | \(-28162115551121937\) | \([2]\) | \(1013760\) | \(1.8440\) |
Rank
sage: E.rank()
The elliptic curves in class 122199.f have rank \(1\).
Complex multiplication
The elliptic curves in class 122199.f do not have complex multiplication.Modular form 122199.2.a.f
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.