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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 605.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
605.b1 | 605b3 | \([1, -1, 1, -7162, -231426]\) | \(22930509321/6875\) | \(12179481875\) | \([2]\) | \(480\) | \(0.91328\) | |
605.b2 | 605b4 | \([1, -1, 1, -3532, 79786]\) | \(2749884201/73205\) | \(129687123005\) | \([2]\) | \(480\) | \(0.91328\) | |
605.b3 | 605b2 | \([1, -1, 1, -507, -2494]\) | \(8120601/3025\) | \(5358972025\) | \([2, 2]\) | \(240\) | \(0.56671\) | |
605.b4 | 605b1 | \([1, -1, 1, 98, -316]\) | \(59319/55\) | \(-97435855\) | \([4]\) | \(120\) | \(0.22013\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 605.b have rank \(1\).
Complex multiplication
The elliptic curves in class 605.b do not have complex multiplication.Modular form 605.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.