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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 405600.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405600.b1 | 405600b2 | \([0, -1, 0, -136608, 19472712]\) | \(7301384/3\) | \(115843416000000\) | \([2]\) | \(2359296\) | \(1.6610\) | \(\Gamma_0(N)\)-optimal* |
405600.b2 | 405600b4 | \([0, -1, 0, -73233, -7461663]\) | \(140608/3\) | \(926747328000000\) | \([2]\) | \(2359296\) | \(1.6610\) | |
405600.b3 | 405600b1 | \([0, -1, 0, -9858, 206712]\) | \(21952/9\) | \(43441281000000\) | \([2, 2]\) | \(1179648\) | \(1.3144\) | \(\Gamma_0(N)\)-optimal* |
405600.b4 | 405600b3 | \([0, -1, 0, 32392, 1474212]\) | \(97336/81\) | \(-3127772232000000\) | \([2]\) | \(2359296\) | \(1.6610\) |
Rank
sage: E.rank()
The elliptic curves in class 405600.b have rank \(1\).
Complex multiplication
The elliptic curves in class 405600.b do not have complex multiplication.Modular form 405600.2.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.