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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 1045.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1045.b1 | 1045b3 | \([1, -1, 0, -1016789, -346894930]\) | \(116256292809537371612841/15216540068579856875\) | \(15216540068579856875\) | \([4]\) | \(17664\) | \(2.4090\) | |
| 1045.b2 | 1045b2 | \([1, -1, 0, -982414, -374539305]\) | \(104859453317683374662841/2223652969140625\) | \(2223652969140625\) | \([2, 2]\) | \(8832\) | \(2.0624\) | |
| 1045.b3 | 1045b1 | \([1, -1, 0, -982409, -374543312]\) | \(104857852278310619039721/47155625\) | \(47155625\) | \([2]\) | \(4416\) | \(1.7158\) | \(\Gamma_0(N)\)-optimal |
| 1045.b4 | 1045b4 | \([1, -1, 0, -948119, -401927292]\) | \(-94256762600623910012361/15323275604248046875\) | \(-15323275604248046875\) | \([2]\) | \(17664\) | \(2.4090\) |
Rank
sage: E.rank()
The elliptic curves in class 1045.b have rank \(1\).
Complex multiplication
The elliptic curves in class 1045.b do not have complex multiplication.Modular form 1045.2.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.
