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SageMath
E = EllipticCurve("ke1")
E.isogeny_class()
Elliptic curves in class 58800.ke
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
58800.ke1 | 58800kf1 | \([0, 1, 0, -14128, 620948]\) | \(5177717/189\) | \(11384658432000\) | \([2]\) | \(147456\) | \(1.2749\) | \(\Gamma_0(N)\)-optimal |
58800.ke2 | 58800kf2 | \([0, 1, 0, 5472, 2228148]\) | \(300763/35721\) | \(-2151700443648000\) | \([2]\) | \(294912\) | \(1.6215\) |
Rank
sage: E.rank()
The elliptic curves in class 58800.ke have rank \(1\).
Complex multiplication
The elliptic curves in class 58800.ke do not have complex multiplication.Modular form 58800.2.a.ke
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.