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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1587.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1587.e1 | 1587c2 | \([1, 0, 1, -8211, 284719]\) | \(413493625/1587\) | \(234932955843\) | \([2]\) | \(2112\) | \(1.0404\) | |
1587.e2 | 1587c1 | \([1, 0, 1, -276, 8581]\) | \(-15625/207\) | \(-30643429023\) | \([2]\) | \(1056\) | \(0.69378\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1587.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1587.e do not have complex multiplication.Modular form 1587.2.a.e
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.