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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 244800.bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
244800.bi1 | 244800bi3 | \([0, 0, 0, -1305900, 574398000]\) | \(82483294977/17\) | \(50761728000000\) | \([2]\) | \(2097152\) | \(2.0171\) | |
244800.bi2 | 244800bi2 | \([0, 0, 0, -81900, 8910000]\) | \(20346417/289\) | \(862949376000000\) | \([2, 2]\) | \(1048576\) | \(1.6705\) | |
244800.bi3 | 244800bi1 | \([0, 0, 0, -9900, -162000]\) | \(35937/17\) | \(50761728000000\) | \([2]\) | \(524288\) | \(1.3240\) | \(\Gamma_0(N)\)-optimal |
244800.bi4 | 244800bi4 | \([0, 0, 0, -9900, 24030000]\) | \(-35937/83521\) | \(-249392369664000000\) | \([2]\) | \(2097152\) | \(2.0171\) |
Rank
sage: E.rank()
The elliptic curves in class 244800.bi have rank \(1\).
Complex multiplication
The elliptic curves in class 244800.bi do not have complex multiplication.Modular form 244800.2.a.bi
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.