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SageMath
E = EllipticCurve("eg1")
E.isogeny_class()
Elliptic curves in class 452400eg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
452400.eg4 | 452400eg1 | \([0, 1, 0, 287592, 66295188]\) | \(41102915774831/53367275520\) | \(-3415505633280000000\) | \([2]\) | \(5529600\) | \(2.2421\) | \(\Gamma_0(N)\)-optimal* |
452400.eg3 | 452400eg2 | \([0, 1, 0, -1760408, 643831188]\) | \(9427227449071249/2652468249600\) | \(169757967974400000000\) | \([2, 2]\) | \(11059200\) | \(2.5887\) | \(\Gamma_0(N)\)-optimal* |
452400.eg1 | 452400eg3 | \([0, 1, 0, -25888408, 50685303188]\) | \(29981943972267024529/4007065140000\) | \(256452168960000000000\) | \([2]\) | \(22118400\) | \(2.9353\) | \(\Gamma_0(N)\)-optimal* |
452400.eg2 | 452400eg4 | \([0, 1, 0, -10400408, -12402568812]\) | \(1943993954077461649/87266819409120\) | \(5585076442183680000000\) | \([2]\) | \(22118400\) | \(2.9353\) |
Rank
sage: E.rank()
The elliptic curves in class 452400eg have rank \(0\).
Complex multiplication
The elliptic curves in class 452400eg do not have complex multiplication.Modular form 452400.2.a.eg
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.