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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 209814g
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 209814.cy3 | 209814g1 | \([1, 0, 0, -289526564, 1706140723728]\) | \(62768149033310713/6915442583808\) | \(295712456581123642732015872\) | \([2]\) | \(132710400\) | \(3.8127\) | \(\Gamma_0(N)\)-optimal |
| 209814.cy2 | 209814g2 | \([1, 0, 0, -1098009844, -12175355497216]\) | \(3423676911662954233/483711578981136\) | \(20684075901687405523301057424\) | \([2, 2]\) | \(265420800\) | \(4.1592\) | |
| 209814.cy4 | 209814g3 | \([1, 0, 0, 1791128936, -65436628906516]\) | \(14861225463775641287/51859390496937804\) | \(-2217568517820699698394138501036\) | \([2]\) | \(530841600\) | \(4.5058\) | |
| 209814.cy1 | 209814g4 | \([1, 0, 0, -16922881104, -847326606295212]\) | \(12534210458299016895673/315581882565708\) | \(13494652383421792645350732972\) | \([2]\) | \(530841600\) | \(4.5058\) |
Rank
sage: E.rank()
The elliptic curves in class 209814g have rank \(0\).
Complex multiplication
The elliptic curves in class 209814g do not have complex multiplication.Modular form 209814.2.a.g
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.
