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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 23104.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
23104.bf1 | 23104bo3 | \([0, 0, 0, -15884, -768208]\) | \(287496\) | \(1541599428608\) | \([2]\) | \(28800\) | \(1.2014\) | \(-16\) | |
23104.bf2 | 23104bo4 | \([0, 0, 0, -15884, 768208]\) | \(287496\) | \(1541599428608\) | \([2]\) | \(28800\) | \(1.2014\) | \(-16\) | |
23104.bf3 | 23104bo2 | \([0, 0, 0, -1444, 0]\) | \(1728\) | \(192699928576\) | \([2, 2]\) | \(14400\) | \(0.85483\) | \(-4\) | |
23104.bf4 | 23104bo1 | \([0, 0, 0, 361, 0]\) | \(1728\) | \(-3010936384\) | \([2]\) | \(7200\) | \(0.50826\) | \(\Gamma_0(N)\)-optimal | \(-4\) |
Rank
sage: E.rank()
The elliptic curves in class 23104.bf have rank \(1\).
Complex multiplication
Each elliptic curve in class 23104.bf has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).Modular form 23104.2.a.bf
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.