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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 55440co
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55440.ew1 | 55440co1 | \([0, 0, 0, -33507, 2182626]\) | \(51603494067/4336640\) | \(349626716651520\) | \([2]\) | \(184320\) | \(1.5324\) | \(\Gamma_0(N)\)-optimal |
55440.ew2 | 55440co2 | \([0, 0, 0, 35613, 10020834]\) | \(61958108493/573927200\) | \(-46270910781849600\) | \([2]\) | \(368640\) | \(1.8790\) |
Rank
sage: E.rank()
The elliptic curves in class 55440co have rank \(1\).
Complex multiplication
The elliptic curves in class 55440co do not have complex multiplication.Modular form 55440.2.a.co
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.