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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 4830.bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4830.bi1 | 4830bh5 | \([1, 0, 0, -84690, -9493338]\) | \(67176973097223766561/91487391870\) | \(91487391870\) | \([2]\) | \(16384\) | \(1.3774\) | |
| 4830.bi2 | 4830bh3 | \([1, 0, 0, -5340, -145908]\) | \(16840406336564161/604708416900\) | \(604708416900\) | \([2, 2]\) | \(8192\) | \(1.0309\) | |
| 4830.bi3 | 4830bh2 | \([1, 0, 0, -840, 6192]\) | \(65553197996161/20996010000\) | \(20996010000\) | \([2, 4]\) | \(4096\) | \(0.68428\) | |
| 4830.bi4 | 4830bh1 | \([1, 0, 0, -760, 8000]\) | \(48551226272641/9273600\) | \(9273600\) | \([4]\) | \(2048\) | \(0.33771\) | \(\Gamma_0(N)\)-optimal |
| 4830.bi5 | 4830bh6 | \([1, 0, 0, 2010, -514878]\) | \(898045580910239/115117148363070\) | \(-115117148363070\) | \([2]\) | \(16384\) | \(1.3774\) | |
| 4830.bi6 | 4830bh4 | \([1, 0, 0, 2380, 42900]\) | \(1490881681033919/1650501562500\) | \(-1650501562500\) | \([4]\) | \(8192\) | \(1.0309\) |
Rank
sage: E.rank()
The elliptic curves in class 4830.bi have rank \(0\).
Complex multiplication
The elliptic curves in class 4830.bi do not have complex multiplication.Modular form 4830.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
