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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 116610.bb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
116610.bb1 | 116610y4 | \([1, 0, 1, -18657604, 31017738806]\) | \(148809678420065817601/20700\) | \(99914946300\) | \([2]\) | \(3538944\) | \(2.4355\) | |
116610.bb2 | 116610y6 | \([1, 0, 1, -4365274, -3007392178]\) | \(1905890658841300321/293666194803750\) | \(1417470632074493733750\) | \([2]\) | \(7077888\) | \(2.7821\) | |
116610.bb3 | 116610y3 | \([1, 0, 1, -1196524, 457952822]\) | \(39248884582600321/3935264062500\) | \(18994767994251562500\) | \([2, 2]\) | \(3538944\) | \(2.4355\) | |
116610.bb4 | 116610y2 | \([1, 0, 1, -1166104, 484576406]\) | \(36330796409313601/428490000\) | \(2068239388410000\) | \([2, 2]\) | \(1769472\) | \(2.0889\) | |
116610.bb5 | 116610y1 | \([1, 0, 1, -70984, 7980182]\) | \(-8194759433281/965779200\) | \(-4661631734572800\) | \([2]\) | \(884736\) | \(1.7423\) | \(\Gamma_0(N)\)-optimal |
116610.bb6 | 116610y5 | \([1, 0, 1, 1485506, 2219510126]\) | \(75108181893694559/484313964843750\) | \(-2337691004333496093750\) | \([2]\) | \(7077888\) | \(2.7821\) |
Rank
sage: E.rank()
The elliptic curves in class 116610.bb have rank \(2\).
Complex multiplication
The elliptic curves in class 116610.bb do not have complex multiplication.Modular form 116610.2.a.bb
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.