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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 82110.bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 82110.bz1 | 82110bx4 | \([1, 0, 0, -275345, -55609533]\) | \(2308635631424282766481/1192703786215830\) | \(1192703786215830\) | \([2]\) | \(1032192\) | \(1.8440\) | |
| 82110.bz2 | 82110bx3 | \([1, 0, 0, -157045, 23564807]\) | \(428347490215633667281/7730096413484970\) | \(7730096413484970\) | \([2]\) | \(1032192\) | \(1.8440\) | |
| 82110.bz3 | 82110bx2 | \([1, 0, 0, -20195, -548163]\) | \(910870482653192881/398111434452900\) | \(398111434452900\) | \([2, 2]\) | \(516096\) | \(1.4975\) | |
| 82110.bz4 | 82110bx1 | \([1, 0, 0, 4305, -63063]\) | \(8823418415295119/6843786390000\) | \(-6843786390000\) | \([4]\) | \(258048\) | \(1.1509\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 82110.bz have rank \(0\).
Complex multiplication
The elliptic curves in class 82110.bz do not have complex multiplication.Modular form 82110.2.a.bz
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.
