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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 198.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
198.a1 | 198a3 | \([1, -1, 0, -3168, 69430]\) | \(4824238966273/66\) | \(48114\) | \([2]\) | \(128\) | \(0.45435\) | |
198.a2 | 198a2 | \([1, -1, 0, -198, 1120]\) | \(1180932193/4356\) | \(3175524\) | \([2, 2]\) | \(64\) | \(0.10778\) | |
198.a3 | 198a4 | \([1, -1, 0, -108, 2074]\) | \(-192100033/2371842\) | \(-1729072818\) | \([2]\) | \(128\) | \(0.45435\) | |
198.a4 | 198a1 | \([1, -1, 0, -18, 4]\) | \(912673/528\) | \(384912\) | \([2]\) | \(32\) | \(-0.23880\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 198.a have rank \(1\).
Complex multiplication
The elliptic curves in class 198.a do not have complex multiplication.Modular form 198.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.