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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 35574.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
35574.q1 | 35574q3 | \([1, 1, 0, -59675508, -177461094180]\) | \(112763292123580561/1932612\) | \(402799592828562468\) | \([2]\) | \(4320000\) | \(2.9198\) | |
35574.q2 | 35574q4 | \([1, 1, 0, -59616218, -177831241650]\) | \(-112427521449300721/466873642818\) | \(-97306915836949221050802\) | \([2]\) | \(8640000\) | \(3.2663\) | |
35574.q3 | 35574q1 | \([1, 1, 0, -266928, 38580480]\) | \(10091699281/2737152\) | \(570483734505366528\) | \([2]\) | \(864000\) | \(2.1150\) | \(\Gamma_0(N)\)-optimal |
35574.q4 | 35574q2 | \([1, 1, 0, 681712, 252024480]\) | \(168105213359/228637728\) | \(-47653219447901397792\) | \([2]\) | \(1728000\) | \(2.4616\) |
Rank
sage: E.rank()
The elliptic curves in class 35574.q have rank \(1\).
Complex multiplication
The elliptic curves in class 35574.q do not have complex multiplication.Modular form 35574.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.