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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 221067.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221067.a1 | 221067b2 | \([0, 0, 1, -2341713, -1379423444]\) | \(-1099616058781696/143578043\) | \(-185426443586204667\) | \([]\) | \(10080000\) | \(2.3338\) | |
221067.a2 | 221067b1 | \([0, 0, 1, 21417, -41594]\) | \(841232384/487403\) | \(-629465362494507\) | \([]\) | \(2016000\) | \(1.5291\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 221067.a have rank \(1\).
Complex multiplication
The elliptic curves in class 221067.a do not have complex multiplication.Modular form 221067.2.a.a
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.