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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 85782.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
85782.m1 | 85782m3 | \([1, 1, 1, -631188, -146187435]\) | \(46753267515625/11591221248\) | \(6894728717181124608\) | \([2]\) | \(1814400\) | \(2.3251\) | |
85782.m2 | 85782m1 | \([1, 1, 1, -214893, 38238819]\) | \(1845026709625/793152\) | \(471785306697792\) | \([2]\) | \(604800\) | \(1.7758\) | \(\Gamma_0(N)\)-optimal |
85782.m3 | 85782m2 | \([1, 1, 1, -181253, 50658707]\) | \(-1107111813625/1228691592\) | \(-730854413238217032\) | \([2]\) | \(1209600\) | \(2.1223\) | |
85782.m4 | 85782m4 | \([1, 1, 1, 1521772, -924697771]\) | \(655215969476375/1001033261568\) | \(-595437929077339427328\) | \([2]\) | \(3628800\) | \(2.6716\) |
Rank
sage: E.rank()
The elliptic curves in class 85782.m have rank \(1\).
Complex multiplication
The elliptic curves in class 85782.m do not have complex multiplication.Modular form 85782.2.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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.