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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 21840.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21840.b1 | 21840x4 | \([0, -1, 0, -410066496, -3196016968704]\) | \(1861772567578966373029167169/9401133413380800000\) | \(38507042461207756800000\) | \([2]\) | \(4423680\) | \(3.5319\) | |
| 21840.b2 | 21840x2 | \([0, -1, 0, -26066496, -48138568704]\) | \(478202393398338853167169/32244226560000000000\) | \(132072351989760000000000\) | \([2, 2]\) | \(2211840\) | \(3.1853\) | |
| 21840.b3 | 21840x1 | \([0, -1, 0, -5094976, 3518479360]\) | \(3571003510905229697089/762141946675200000\) | \(3121733413581619200000\) | \([2]\) | \(1105920\) | \(2.8387\) | \(\Gamma_0(N)\)-optimal |
| 21840.b4 | 21840x3 | \([0, -1, 0, 22389184, -206646789120]\) | \(303025056761573589385151/4678857421875000000000\) | \(-19164600000000000000000000\) | \([2]\) | \(4423680\) | \(3.5319\) |
Rank
sage: E.rank()
The elliptic curves in class 21840.b have rank \(0\).
Complex multiplication
The elliptic curves in class 21840.b do not have complex multiplication.Modular form 21840.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
