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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 349830j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
349830.j3 | 349830j1 | \([1, -1, 0, -1018730595, 12524687954325]\) | \(-897176485088045307663363/767948742459392000\) | \(-100082031344325236883456000\) | \([2]\) | \(174182400\) | \(3.9159\) | \(\Gamma_0(N)\)-optimal |
349830.j2 | 349830j2 | \([1, -1, 0, -16303036515, 801222385176981]\) | \(3677099129012869569042846723/2690674688000000\) | \(350659065602985984000000\) | \([2]\) | \(348364800\) | \(4.2624\) | |
349830.j4 | 349830j3 | \([1, -1, 0, 1109858205, 54915589284245]\) | \(1591383301847324275653/14633798888411156480\) | \(-1390299890534788285803457474560\) | \([2]\) | \(522547200\) | \(4.4652\) | |
349830.j1 | 349830j4 | \([1, -1, 0, -16655908515, 764725479661781]\) | \(5378699555702101965641787/453548482696362123200\) | \(43089864132568698316981260590400\) | \([2]\) | \(1045094400\) | \(4.8117\) |
Rank
sage: E.rank()
The elliptic curves in class 349830j have rank \(1\).
Complex multiplication
The elliptic curves in class 349830j do not have complex multiplication.Modular form 349830.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.