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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 117.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
117.a1 | 117a4 | \([1, -1, 1, -626, 6180]\) | \(37159393753/1053\) | \(767637\) | \([2]\) | \(32\) | \(0.23094\) | |
117.a2 | 117a3 | \([1, -1, 1, -176, -768]\) | \(822656953/85683\) | \(62462907\) | \([2]\) | \(32\) | \(0.23094\) | |
117.a3 | 117a2 | \([1, -1, 1, -41, 96]\) | \(10218313/1521\) | \(1108809\) | \([2, 2]\) | \(16\) | \(-0.11563\) | |
117.a4 | 117a1 | \([1, -1, 1, 4, 6]\) | \(12167/39\) | \(-28431\) | \([4]\) | \(8\) | \(-0.46221\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 117.a have rank \(1\).
Complex multiplication
The elliptic curves in class 117.a do not have complex multiplication.Modular form 117.2.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.