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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 285318.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
285318.x1 | 285318x1 | \([1, -1, 1, -77342, -8218515]\) | \(39616946929/226368\) | \(292347021206592\) | \([2]\) | \(3317760\) | \(1.6172\) | \(\Gamma_0(N)\)-optimal |
285318.x2 | 285318x2 | \([1, -1, 1, -33782, -17453235]\) | \(-3301293169/100082952\) | \(-129253926750964488\) | \([2]\) | \(6635520\) | \(1.9638\) |
Rank
sage: E.rank()
The elliptic curves in class 285318.x have rank \(2\).
Complex multiplication
The elliptic curves in class 285318.x do not have complex multiplication.Modular form 285318.2.a.x
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.