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SageMath
E = EllipticCurve("gy1")
E.isogeny_class()
Elliptic curves in class 426888gy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
426888.gy4 | 426888gy1 | \([0, 0, 0, 1476321, 27330804578]\) | \(9148592/8301447\) | \(-322898214978979871438592\) | \([2]\) | \(47185920\) | \(3.1900\) | \(\Gamma_0(N)\)-optimal* |
426888.gy3 | 426888gy2 | \([0, 0, 0, -127657299, 542393161310]\) | \(1478729816932/38900169\) | \(6052340095639060730766336\) | \([2, 2]\) | \(94371840\) | \(3.5366\) | \(\Gamma_0(N)\)-optimal* |
426888.gy1 | 426888gy3 | \([0, 0, 0, -2029443339, 35189511595238]\) | \(2970658109581346/2139291\) | \(665689483022029126391808\) | \([2]\) | \(188743680\) | \(3.8831\) | \(\Gamma_0(N)\)-optimal* |
426888.gy2 | 426888gy4 | \([0, 0, 0, -292009179, -1140734441770]\) | \(8849350367426/3314597517\) | \(1031413074480204609209296896\) | \([2]\) | \(188743680\) | \(3.8831\) |
Rank
sage: E.rank()
The elliptic curves in class 426888gy have rank \(1\).
Complex multiplication
The elliptic curves in class 426888gy do not have complex multiplication.Modular form 426888.2.a.gy
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 Cremona 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 Cremona labels.