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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 410.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
410.a1 | 410c4 | \([1, 0, 1, -3143, 32586]\) | \(3432086371273321/1520033357120\) | \(1520033357120\) | \([2]\) | \(768\) | \(1.0332\) | |
410.a2 | 410c2 | \([1, 0, 1, -2668, 52806]\) | \(2099167877572921/840500\) | \(840500\) | \([6]\) | \(256\) | \(0.48385\) | |
410.a3 | 410c3 | \([1, 0, 1, -1543, -23094]\) | \(405897921250921/7057510400\) | \(7057510400\) | \([2]\) | \(384\) | \(0.68658\) | |
410.a4 | 410c1 | \([1, 0, 1, -168, 806]\) | \(519912412921/10250000\) | \(10250000\) | \([6]\) | \(128\) | \(0.13728\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 410.a have rank \(0\).
Complex multiplication
The elliptic curves in class 410.a do not have complex multiplication.Modular form 410.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.