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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 8190.bz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8190.bz1 | 8190by3 | \([1, -1, 1, -70592, -7201371]\) | \(53365044437418169/41984670\) | \(30606824430\) | \([2]\) | \(24576\) | \(1.3186\) | |
8190.bz2 | 8190by4 | \([1, -1, 1, -10292, 245589]\) | \(165369706597369/60703354530\) | \(44252745452370\) | \([2]\) | \(24576\) | \(1.3186\) | |
8190.bz3 | 8190by2 | \([1, -1, 1, -4442, -110091]\) | \(13293525831769/365192100\) | \(266225040900\) | \([2, 2]\) | \(12288\) | \(0.97207\) | |
8190.bz4 | 8190by1 | \([1, -1, 1, 58, -5691]\) | \(30080231/19110000\) | \(-13931190000\) | \([4]\) | \(6144\) | \(0.62550\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8190.bz have rank \(0\).
Complex multiplication
The elliptic curves in class 8190.bz do not have complex multiplication.Modular form 8190.2.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.