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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1600.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1600.k1 | 1600p3 | \([0, 0, 0, -10700, 426000]\) | \(132304644/5\) | \(5120000000\) | \([2]\) | \(1536\) | \(0.94908\) | |
1600.k2 | 1600p2 | \([0, 0, 0, -700, 6000]\) | \(148176/25\) | \(6400000000\) | \([2, 2]\) | \(768\) | \(0.60250\) | |
1600.k3 | 1600p1 | \([0, 0, 0, -200, -1000]\) | \(55296/5\) | \(80000000\) | \([2]\) | \(384\) | \(0.25593\) | \(\Gamma_0(N)\)-optimal |
1600.k4 | 1600p4 | \([0, 0, 0, 1300, 34000]\) | \(237276/625\) | \(-640000000000\) | \([2]\) | \(1536\) | \(0.94908\) |
Rank
sage: E.rank()
The elliptic curves in class 1600.k have rank \(0\).
Complex multiplication
The elliptic curves in class 1600.k do not have complex multiplication.Modular form 1600.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.