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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 213444ch
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
213444.dm2 | 213444ch1 | \([0, 0, 0, -4126584, -3750418595]\) | \(-3196715008/649539\) | \(-1579054408069096317744\) | \([2]\) | \(10368000\) | \(2.7900\) | \(\Gamma_0(N)\)-optimal |
213444.dm1 | 213444ch2 | \([0, 0, 0, -68960199, -220411393202]\) | \(932410994128/29403\) | \(1143677266749633547008\) | \([2]\) | \(20736000\) | \(3.1366\) |
Rank
sage: E.rank()
The elliptic curves in class 213444ch have rank \(1\).
Complex multiplication
The elliptic curves in class 213444ch do not have complex multiplication.Modular form 213444.2.a.ch
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.