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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 27200.bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27200.bi1 | 27200c4 | \([0, 0, 0, -145100, -21274000]\) | \(82483294977/17\) | \(69632000000\) | \([2]\) | \(65536\) | \(1.4678\) | |
| 27200.bi2 | 27200c2 | \([0, 0, 0, -9100, -330000]\) | \(20346417/289\) | \(1183744000000\) | \([2, 2]\) | \(32768\) | \(1.1212\) | |
| 27200.bi3 | 27200c3 | \([0, 0, 0, -1100, -890000]\) | \(-35937/83521\) | \(-342102016000000\) | \([2]\) | \(65536\) | \(1.4678\) | |
| 27200.bi4 | 27200c1 | \([0, 0, 0, -1100, 6000]\) | \(35937/17\) | \(69632000000\) | \([2]\) | \(16384\) | \(0.77466\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 27200.bi have rank \(1\).
Complex multiplication
The elliptic curves in class 27200.bi do not have complex multiplication.Modular form 27200.2.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.
