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SageMath
E = EllipticCurve("hb1")
E.isogeny_class()
Elliptic curves in class 476850hb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
476850.hb4 | 476850hb1 | \([1, 1, 1, -25630838, -8203683469]\) | \(4937402992298041/2780405760000\) | \(1048628685624960000000000\) | \([2]\) | \(63700992\) | \(3.2992\) | \(\Gamma_0(N)\)-optimal* |
476850.hb2 | 476850hb2 | \([1, 1, 1, -256830838, 1575978716531]\) | \(4967657717692586041/29490113030400\) | \(11122181845141860900000000\) | \([2, 2]\) | \(127401984\) | \(3.6458\) | \(\Gamma_0(N)\)-optimal* |
476850.hb1 | 476850hb3 | \([1, 1, 1, -4103420838, 101171886996531]\) | \(20260414982443110947641/720358602480\) | \(271682898001633923750000\) | \([2]\) | \(254803968\) | \(3.9924\) | \(\Gamma_0(N)\)-optimal* |
476850.hb3 | 476850hb4 | \([1, 1, 1, -109440838, 3372368036531]\) | \(-384369029857072441/12804787777021680\) | \(-4829319507784647117123750000\) | \([2]\) | \(254803968\) | \(3.9924\) |
Rank
sage: E.rank()
The elliptic curves in class 476850hb have rank \(1\).
Complex multiplication
The elliptic curves in class 476850hb do not have complex multiplication.Modular form 476850.2.a.hb
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.