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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 40560.cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
40560.cf1 | 40560x2 | \([0, 1, 0, -11539376, 15071501940]\) | \(7824392006186/7381125\) | \(160303381369926912000\) | \([2]\) | \(2995200\) | \(2.7991\) | |
40560.cf2 | 40560x1 | \([0, 1, 0, -554376, 347207940]\) | \(-1735192372/3796875\) | \(-41230293562224000000\) | \([2]\) | \(1497600\) | \(2.4526\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 40560.cf have rank \(1\).
Complex multiplication
The elliptic curves in class 40560.cf do not have complex multiplication.Modular form 40560.2.a.cf
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.