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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 468270k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
468270.k4 | 468270k1 | \([1, -1, 0, -1325880, 3184645360]\) | \(-199596497460121/3276305374320\) | \(-4231243447596835156080\) | \([2]\) | \(37601280\) | \(2.8310\) | \(\Gamma_0(N)\)-optimal* |
468270.k3 | 468270k2 | \([1, -1, 0, -41597100, 102888131836]\) | \(6163526129192423641/26532491940900\) | \(34265863479422991032100\) | \([2, 2]\) | \(75202560\) | \(3.1776\) | \(\Gamma_0(N)\)-optimal* |
468270.k1 | 468270k3 | \([1, -1, 0, -664864470, 6598705315450]\) | \(25167463564736957591161/469382141250\) | \(606192000645008621250\) | \([2]\) | \(150405120\) | \(3.5241\) | \(\Gamma_0(N)\)-optimal* |
468270.k2 | 468270k4 | \([1, -1, 0, -62669250, -12389171954]\) | \(21076746329185034041/12145717817910270\) | \(15685805522343688221141630\) | \([2]\) | \(150405120\) | \(3.5241\) |
Rank
sage: E.rank()
The elliptic curves in class 468270k have rank \(0\).
Complex multiplication
The elliptic curves in class 468270k do not have complex multiplication.Modular form 468270.2.a.k
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.