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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 201810ce
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
201810.n2 | 201810ce1 | \([1, 1, 0, -289947, 59936481]\) | \(2805165723497909881/1953492187500\) | \(1877305992187500\) | \([]\) | \(2643840\) | \(1.8674\) | \(\Gamma_0(N)\)-optimal |
201810.n1 | 201810ce2 | \([1, 1, 0, -23481822, 43787362356]\) | \(1490032455664120989539881/504000\) | \(484344000\) | \([]\) | \(7931520\) | \(2.4167\) |
Rank
sage: E.rank()
The elliptic curves in class 201810ce have rank \(2\).
Complex multiplication
The elliptic curves in class 201810ce do not have complex multiplication.Modular form 201810.2.a.ce
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.