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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 166410.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
166410.q1 | 166410cj4 | \([1, -1, 0, -402504, 25793810]\) | \(57960603/31250\) | \(3888230902920843750\) | \([2]\) | \(3773952\) | \(2.2581\) | |
166410.q2 | 166410cj2 | \([1, -1, 0, -236094, -44094692]\) | \(8527173507/200\) | \(34135360464600\) | \([2]\) | \(1257984\) | \(1.7088\) | |
166410.q3 | 166410cj1 | \([1, -1, 0, -14214, -739340]\) | \(-1860867/320\) | \(-54616576743360\) | \([2]\) | \(628992\) | \(1.3622\) | \(\Gamma_0(N)\)-optimal |
166410.q4 | 166410cj3 | \([1, -1, 0, 96726, 3128768]\) | \(804357/500\) | \(-62211694446733500\) | \([2]\) | \(1886976\) | \(1.9115\) |
Rank
sage: E.rank()
The elliptic curves in class 166410.q have rank \(1\).
Complex multiplication
The elliptic curves in class 166410.q do not have complex multiplication.Modular form 166410.2.a.q
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.