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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 7200.bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
7200.bg1 | 7200g3 | \([0, 0, 0, -18300, 952000]\) | \(14526784/15\) | \(699840000000\) | \([2]\) | \(12288\) | \(1.1908\) | |
7200.bg2 | 7200g2 | \([0, 0, 0, -12675, -544250]\) | \(38614472/405\) | \(2361960000000\) | \([2]\) | \(12288\) | \(1.1908\) | |
7200.bg3 | 7200g1 | \([0, 0, 0, -1425, 7000]\) | \(438976/225\) | \(164025000000\) | \([2, 2]\) | \(6144\) | \(0.84418\) | \(\Gamma_0(N)\)-optimal |
7200.bg4 | 7200g4 | \([0, 0, 0, 5325, 54250]\) | \(2863288/1875\) | \(-10935000000000\) | \([2]\) | \(12288\) | \(1.1908\) |
Rank
sage: E.rank()
The elliptic curves in class 7200.bg have rank \(0\).
Complex multiplication
The elliptic curves in class 7200.bg do not have complex multiplication.Modular form 7200.2.a.bg
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.