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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 30960.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30960.bf1 | 30960bw3 | \([0, 0, 0, -4940067, -1835863774]\) | \(4465136636671380769/2096375976562500\) | \(6259745124000000000000\) | \([2]\) | \(1658880\) | \(2.8767\) | |
30960.bf2 | 30960bw1 | \([0, 0, 0, -2529507, 1548391394]\) | \(599437478278595809/33854760000\) | \(101089771683840000\) | \([2]\) | \(552960\) | \(2.3274\) | \(\Gamma_0(N)\)-optimal |
30960.bf3 | 30960bw2 | \([0, 0, 0, -2385507, 1732452194]\) | \(-502780379797811809/143268096832200\) | \(-427796244851399884800\) | \([2]\) | \(1105920\) | \(2.6739\) | |
30960.bf4 | 30960bw4 | \([0, 0, 0, 17559933, -13900363774]\) | \(200541749524551119231/144008551960031250\) | \(-430007232015821952000000\) | \([2]\) | \(3317760\) | \(3.2232\) |
Rank
sage: E.rank()
The elliptic curves in class 30960.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 30960.bf do not have complex multiplication.Modular form 30960.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.