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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 1710.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1710.m1 | 1710l2 | \([1, -1, 1, -3593, -45143]\) | \(260549802603/104256800\) | \(2052086594400\) | \([2]\) | \(3840\) | \(1.0606\) | |
1710.m2 | 1710l1 | \([1, -1, 1, 727, -5399]\) | \(2161700757/1848320\) | \(-36380482560\) | \([2]\) | \(1920\) | \(0.71404\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1710.m have rank \(1\).
Complex multiplication
The elliptic curves in class 1710.m do not have complex multiplication.Modular form 1710.2.a.m
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.