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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 5586.y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5586.y1 | 5586bb3 | \([1, 0, 0, -4290049, 3419763689]\) | \(74220219816682217473/16416\) | \(1931325984\) | \([2]\) | \(92160\) | \(2.0744\) | |
5586.y2 | 5586bb2 | \([1, 0, 0, -268129, 53416649]\) | \(18120364883707393/269485056\) | \(31704647353344\) | \([2, 2]\) | \(46080\) | \(1.7278\) | |
5586.y3 | 5586bb4 | \([1, 0, 0, -260289, 56689065]\) | \(-16576888679672833/2216253521952\) | \(-260740010604130848\) | \([2]\) | \(92160\) | \(2.0744\) | |
5586.y4 | 5586bb1 | \([1, 0, 0, -17249, 782025]\) | \(4824238966273/537919488\) | \(63285689843712\) | \([2]\) | \(23040\) | \(1.3812\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5586.y have rank \(1\).
Complex multiplication
The elliptic curves in class 5586.y do not have complex multiplication.Modular form 5586.2.a.y
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.