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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 2070.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2070.b1 | 2070f3 | \([1, -1, 0, -993600, -380962764]\) | \(148809678420065817601/20700\) | \(15090300\) | \([2]\) | \(12288\) | \(1.7023\) | |
2070.b2 | 2070f5 | \([1, -1, 0, -232470, 37012950]\) | \(1905890658841300321/293666194803750\) | \(214082656011933750\) | \([2]\) | \(24576\) | \(2.0489\) | |
2070.b3 | 2070f4 | \([1, -1, 0, -63720, -5613300]\) | \(39248884582600321/3935264062500\) | \(2868807501562500\) | \([2, 2]\) | \(12288\) | \(1.7023\) | |
2070.b4 | 2070f2 | \([1, -1, 0, -62100, -5940864]\) | \(36330796409313601/428490000\) | \(312369210000\) | \([2, 2]\) | \(6144\) | \(1.3557\) | |
2070.b5 | 2070f1 | \([1, -1, 0, -3780, -97200]\) | \(-8194759433281/965779200\) | \(-704053036800\) | \([2]\) | \(3072\) | \(1.0092\) | \(\Gamma_0(N)\)-optimal |
2070.b6 | 2070f6 | \([1, -1, 0, 79110, -27294894]\) | \(75108181893694559/484313964843750\) | \(-353064880371093750\) | \([2]\) | \(24576\) | \(2.0489\) |
Rank
sage: E.rank()
The elliptic curves in class 2070.b have rank \(1\).
Complex multiplication
The elliptic curves in class 2070.b do not have complex multiplication.Modular form 2070.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.