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SageMath
E = EllipticCurve("lz1")
E.isogeny_class()
Elliptic curves in class 487872.lz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
487872.lz1 | 487872lz3 | \([0, 0, 0, -2788852044, -40195029294992]\) | \(14171198121996897746/4077720290568771\) | \(690257283227443598222439088128\) | \([2]\) | \(707788800\) | \(4.4320\) | \(\Gamma_0(N)\)-optimal* |
487872.lz2 | 487872lz2 | \([0, 0, 0, -2556938604, -49759417856720]\) | \(21843440425782779332/3100814593569\) | \(262445644211953132592234496\) | \([2, 2]\) | \(353894400\) | \(4.0854\) | \(\Gamma_0(N)\)-optimal* |
487872.lz3 | 487872lz1 | \([0, 0, 0, -2556851484, -49762978590512]\) | \(87364831012240243408/1760913\) | \(37259882261638299648\) | \([2]\) | \(176947200\) | \(3.7388\) | \(\Gamma_0(N)\)-optimal* |
487872.lz4 | 487872lz4 | \([0, 0, 0, -2326419084, -59095919455760]\) | \(-8226100326647904626/4152140742401883\) | \(-702854826766095092257341702144\) | \([2]\) | \(707788800\) | \(4.4320\) |
Rank
sage: E.rank()
The elliptic curves in class 487872.lz have rank \(1\).
Complex multiplication
The elliptic curves in class 487872.lz do not have complex multiplication.Modular form 487872.2.a.lz
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 LMFDB 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 LMFDB labels.