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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 1600q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1600.i3 | 1600q1 | \([0, -1, 0, -193, -1183]\) | \(-121945/32\) | \(-209715200\) | \([]\) | \(384\) | \(0.31362\) | \(\Gamma_0(N)\)-optimal |
1600.i4 | 1600q2 | \([0, -1, 0, 1407, 8737]\) | \(46969655/32768\) | \(-214748364800\) | \([]\) | \(1152\) | \(0.86293\) | |
1600.i2 | 1600q3 | \([0, -1, 0, -833, 109537]\) | \(-25/2\) | \(-5120000000000\) | \([]\) | \(1920\) | \(1.1183\) | |
1600.i1 | 1600q4 | \([0, -1, 0, -200833, 34709537]\) | \(-349938025/8\) | \(-20480000000000\) | \([]\) | \(5760\) | \(1.6676\) |
Rank
sage: E.rank()
The elliptic curves in class 1600q have rank \(0\).
Complex multiplication
The elliptic curves in class 1600q do not have complex multiplication.Modular form 1600.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 5 & 15 \\ 3 & 1 & 15 & 5 \\ 5 & 15 & 1 & 3 \\ 15 & 5 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.