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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 308550ec
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
308550.ec4 | 308550ec1 | \([1, 0, 1, -10731251, 2218704398]\) | \(4937402992298041/2780405760000\) | \(76963412634240000000000\) | \([2]\) | \(26542080\) | \(3.0816\) | \(\Gamma_0(N)\)-optimal |
308550.ec2 | 308550ec2 | \([1, 0, 1, -107531251, -426992495602]\) | \(4967657717692586041/29490113030400\) | \(816305220785132100000000\) | \([2, 2]\) | \(53084160\) | \(3.4281\) | |
308550.ec3 | 308550ec3 | \([1, 0, 1, -45821251, -913637555602]\) | \(-384369029857072441/12804787777021680\) | \(-354444728735129756913750000\) | \([2]\) | \(106168320\) | \(3.7747\) | |
308550.ec1 | 308550ec4 | \([1, 0, 1, -1718041251, -27409477035602]\) | \(20260414982443110947641/720358602480\) | \(19939987596376113750000\) | \([2]\) | \(106168320\) | \(3.7747\) |
Rank
sage: E.rank()
The elliptic curves in class 308550ec have rank \(1\).
Complex multiplication
The elliptic curves in class 308550ec do not have complex multiplication.Modular form 308550.2.a.ec
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.