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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 2005.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2005.b1 | 2005a3 | \([1, -1, 1, -53467, -4745166]\) | \(16903379714178579201/10025\) | \(10025\) | \([2]\) | \(1792\) | \(0.99202\) | |
| 2005.b2 | 2005a2 | \([1, -1, 1, -3342, -73516]\) | \(4126874048961201/100500625\) | \(100500625\) | \([2, 2]\) | \(896\) | \(0.64544\) | |
| 2005.b3 | 2005a4 | \([1, -1, 1, -3217, -79366]\) | \(-3680868702543201/646424040025\) | \(-646424040025\) | \([4]\) | \(1792\) | \(0.99202\) | |
| 2005.b4 | 2005a1 | \([1, -1, 1, -217, -1016]\) | \(1125188511201/156640625\) | \(156640625\) | \([4]\) | \(448\) | \(0.29887\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2005.b have rank \(1\).
Complex multiplication
The elliptic curves in class 2005.b do not have complex multiplication.Modular form 2005.2.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.
