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SageMath
E = EllipticCurve("sm1")
E.isogeny_class()
Elliptic curves in class 310464.sm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
310464.sm1 | 310464sm2 | \([0, 0, 0, -129948, -15448720]\) | \(4662947952/717409\) | \(37336943639052288\) | \([2]\) | \(2949120\) | \(1.9034\) | |
310464.sm2 | 310464sm1 | \([0, 0, 0, 14112, -1330840]\) | \(95551488/290521\) | \(-944994957806592\) | \([2]\) | \(1474560\) | \(1.5568\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 310464.sm have rank \(1\).
Complex multiplication
The elliptic curves in class 310464.sm do not have complex multiplication.Modular form 310464.2.a.sm
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.