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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 110466bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
110466.bi4 | 110466bp1 | \([1, -1, 1, -9815, 234159]\) | \(3048625/1088\) | \(37314534606912\) | \([2]\) | \(331776\) | \(1.3051\) | \(\Gamma_0(N)\)-optimal |
110466.bi3 | 110466bp2 | \([1, -1, 1, -139775, 20144031]\) | \(8805624625/2312\) | \(79293386039688\) | \([2]\) | \(663552\) | \(1.6517\) | |
110466.bi2 | 110466bp3 | \([1, -1, 1, -334715, -74440857]\) | \(120920208625/19652\) | \(673993781337348\) | \([2]\) | \(995328\) | \(1.8544\) | |
110466.bi1 | 110466bp4 | \([1, -1, 1, -367205, -59092581]\) | \(159661140625/48275138\) | \(1655665723855195362\) | \([2]\) | \(1990656\) | \(2.2010\) |
Rank
sage: E.rank()
The elliptic curves in class 110466bp have rank \(1\).
Complex multiplication
The elliptic curves in class 110466bp do not have complex multiplication.Modular form 110466.2.a.bp
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.