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SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 221760ed
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221760.ej4 | 221760ed1 | \([0, 0, 0, 13092, 552368]\) | \(20777545136/23059575\) | \(-275422087987200\) | \([2]\) | \(524288\) | \(1.4573\) | \(\Gamma_0(N)\)-optimal |
221760.ej3 | 221760ed2 | \([0, 0, 0, -74028, 5187152]\) | \(939083699236/300155625\) | \(14340158300160000\) | \([2, 2]\) | \(1048576\) | \(1.8039\) | |
221760.ej1 | 221760ed3 | \([0, 0, 0, -1071948, 427107728]\) | \(1425631925916578/270703125\) | \(25866086400000000\) | \([2]\) | \(2097152\) | \(2.1504\) | |
221760.ej2 | 221760ed4 | \([0, 0, 0, -470028, -120107248]\) | \(120186986927618/4332064275\) | \(413935187587891200\) | \([2]\) | \(2097152\) | \(2.1504\) |
Rank
sage: E.rank()
The elliptic curves in class 221760ed have rank \(0\).
Complex multiplication
The elliptic curves in class 221760ed do not have complex multiplication.Modular form 221760.2.a.ed
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.