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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 800.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
800.f1 | 800d1 | \([0, 0, 0, -125, 0]\) | \(1728\) | \(125000000\) | \([2]\) | \(160\) | \(0.24312\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
800.f2 | 800d2 | \([0, 0, 0, 500, 0]\) | \(1728\) | \(-8000000000\) | \([2]\) | \(320\) | \(0.58969\) | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 800.f have rank \(0\).
Complex multiplication
Each elliptic curve in class 800.f has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 800.2.a.f
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.