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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 200c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
200.c3 | 200c1 | \([0, 0, 0, -50, 125]\) | \(55296/5\) | \(1250000\) | \([4]\) | \(24\) | \(-0.090642\) | \(\Gamma_0(N)\)-optimal |
200.c2 | 200c2 | \([0, 0, 0, -175, -750]\) | \(148176/25\) | \(100000000\) | \([2, 2]\) | \(48\) | \(0.25593\) | |
200.c1 | 200c3 | \([0, 0, 0, -2675, -53250]\) | \(132304644/5\) | \(80000000\) | \([2]\) | \(96\) | \(0.60250\) | |
200.c4 | 200c4 | \([0, 0, 0, 325, -4250]\) | \(237276/625\) | \(-10000000000\) | \([2]\) | \(96\) | \(0.60250\) |
Rank
sage: E.rank()
The elliptic curves in class 200c have rank \(0\).
Complex multiplication
The elliptic curves in class 200c do not have complex multiplication.Modular form 200.2.a.c
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.