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SageMath
E = EllipticCurve("fc1")
E.isogeny_class()
Elliptic curves in class 28224.fc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
28224.fc1 | 28224fp3 | \([0, 0, 0, -19404, -1037232]\) | \(287496\) | \(2810384252928\) | \([2]\) | \(49152\) | \(1.2514\) | \(-16\) | |
28224.fc2 | 28224fp4 | \([0, 0, 0, -19404, 1037232]\) | \(287496\) | \(2810384252928\) | \([2]\) | \(49152\) | \(1.2514\) | \(-16\) | |
28224.fc3 | 28224fp2 | \([0, 0, 0, -1764, 0]\) | \(1728\) | \(351298031616\) | \([2, 2]\) | \(24576\) | \(0.90488\) | \(-4\) | |
28224.fc4 | 28224fp1 | \([0, 0, 0, 441, 0]\) | \(1728\) | \(-5489031744\) | \([2]\) | \(12288\) | \(0.55830\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 28224.fc have rank \(0\).
Complex multiplication
Each elliptic curve in class 28224.fc has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 28224.2.a.fc
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.