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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 3872.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
3872.f1 | 3872b2 | \([0, 0, 0, -1331, -18634]\) | \(287496\) | \(907039232\) | \([2]\) | \(1440\) | \(0.58156\) | \(-16\) | |
3872.f2 | 3872b3 | \([0, 0, 0, -1331, 18634]\) | \(287496\) | \(907039232\) | \([2]\) | \(1440\) | \(0.58156\) | \(-16\) | |
3872.f3 | 3872b1 | \([0, 0, 0, -121, 0]\) | \(1728\) | \(113379904\) | \([2, 2]\) | \(720\) | \(0.23499\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
3872.f4 | 3872b4 | \([0, 0, 0, 484, 0]\) | \(1728\) | \(-7256313856\) | \([2]\) | \(1440\) | \(0.58156\) | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 3872.f have rank \(0\).
Complex multiplication
Each elliptic curve in class 3872.f has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 3872.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.