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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 55.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55.a1 | 55a3 | \([1, -1, 0, -59, 190]\) | \(22930509321/6875\) | \(6875\) | \([4]\) | \(4\) | \(-0.28567\) | |
55.a2 | 55a2 | \([1, -1, 0, -29, -52]\) | \(2749884201/73205\) | \(73205\) | \([2]\) | \(4\) | \(-0.28567\) | |
55.a3 | 55a1 | \([1, -1, 0, -4, 3]\) | \(8120601/3025\) | \(3025\) | \([2, 2]\) | \(2\) | \(-0.63224\) | \(\Gamma_0(N)\)-optimal |
55.a4 | 55a4 | \([1, -1, 0, 1, 0]\) | \(59319/55\) | \(-55\) | \([2]\) | \(4\) | \(-0.97881\) |
Rank
sage: E.rank()
The elliptic curves in class 55.a have rank \(0\).
Complex multiplication
The elliptic curves in class 55.a do not have complex multiplication.Modular form 55.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.