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SageMath
E = EllipticCurve("dc1")
E.isogeny_class()
Elliptic curves in class 152592dc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
152592.g3 | 152592dc1 | \([0, -1, 0, -8268964, -9149428640]\) | \(10119139303540048/85833\) | \(530380789754112\) | \([2]\) | \(3538944\) | \(2.4117\) | \(\Gamma_0(N)\)-optimal |
152592.g2 | 152592dc2 | \([0, -1, 0, -8274744, -9135991296]\) | \(2535093488117092/7367303889\) | \(182096697307858781184\) | \([2, 2]\) | \(7077888\) | \(2.7582\) | |
152592.g1 | 152592dc3 | \([0, -1, 0, -11708064, -827356896]\) | \(3590504967602306/2071799959977\) | \(102416823271714971469824\) | \([2]\) | \(14155776\) | \(3.1048\) | |
152592.g4 | 152592dc4 | \([0, -1, 0, -4933904, -16584728160]\) | \(-268702931670626/2248659199809\) | \(-111159637182207025809408\) | \([2]\) | \(14155776\) | \(3.1048\) |
Rank
sage: E.rank()
The elliptic curves in class 152592dc have rank \(1\).
Complex multiplication
The elliptic curves in class 152592dc do not have complex multiplication.Modular form 152592.2.a.dc
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.