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SageMath
E = EllipticCurve("ls1")
E.isogeny_class()
Elliptic curves in class 411840ls
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
411840.ls4 | 411840ls1 | \([0, 0, 0, -344172, 894329264]\) | \(-23592983745241/1794399750000\) | \(-342915132358656000000\) | \([2]\) | \(10616832\) | \(2.6200\) | \(\Gamma_0(N)\)-optimal* |
411840.ls3 | 411840ls2 | \([0, 0, 0, -16184172, 24888761264]\) | \(2453170411237305241/19353090685500\) | \(3698433224796930048000\) | \([2]\) | \(21233664\) | \(2.9666\) | \(\Gamma_0(N)\)-optimal* |
411840.ls2 | 411840ls3 | \([0, 0, 0, -81776172, 284640482864]\) | \(-316472948332146183241/7074906009600\) | \(-1352035593354844569600\) | \([2]\) | \(31850496\) | \(3.1693\) | \(\Gamma_0(N)\)-optimal* |
411840.ls1 | 411840ls4 | \([0, 0, 0, -1308425772, 18216785655344]\) | \(1296294060988412126189641/647824320\) | \(123801155477176320\) | \([2]\) | \(63700992\) | \(3.5159\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 411840ls have rank \(1\).
Complex multiplication
The elliptic curves in class 411840ls do not have complex multiplication.Modular form 411840.2.a.ls
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.