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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 805.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
805.b1 | 805b1 | \([1, -1, 1, -163, -758]\) | \(476196576129/197225\) | \(197225\) | \([2]\) | \(144\) | \(-0.022016\) | \(\Gamma_0(N)\)-optimal |
805.b2 | 805b2 | \([1, -1, 1, -138, -1018]\) | \(-288673724529/311181605\) | \(-311181605\) | \([2]\) | \(288\) | \(0.32456\) |
Rank
sage: E.rank()
The elliptic curves in class 805.b have rank \(0\).
Complex multiplication
The elliptic curves in class 805.b do not have complex multiplication.Modular form 805.2.a.b
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.