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SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 223440de
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
223440.ck2 | 223440de1 | \([0, -1, 0, -61560, -20600208]\) | \(-53540005609/350208000\) | \(-168761839583232000\) | \([2]\) | \(2903040\) | \(1.9890\) | \(\Gamma_0(N)\)-optimal |
223440.ck1 | 223440de2 | \([0, -1, 0, -1566840, -752768400]\) | \(882774443450089/2166000000\) | \(1043774398464000000\) | \([2]\) | \(5806080\) | \(2.3356\) |
Rank
sage: E.rank()
The elliptic curves in class 223440de have rank \(1\).
Complex multiplication
The elliptic curves in class 223440de do not have complex multiplication.Modular form 223440.2.a.de
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.