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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 27209.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27209.m1 | 27209e2 | \([1, 1, 0, -16565, -567094]\) | \(104154702625/32188247\) | \(155366520313823\) | \([2]\) | \(80640\) | \(1.4276\) | |
27209.m2 | 27209e1 | \([1, 1, 0, 2870, -57897]\) | \(541343375/625807\) | \(-3020650859863\) | \([2]\) | \(40320\) | \(1.0811\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 27209.m have rank \(1\).
Complex multiplication
The elliptic curves in class 27209.m do not have complex multiplication.Modular form 27209.2.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.