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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 27209a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27209.j1 | 27209a1 | \([0, 1, 1, -7383159, 7719213259]\) | \(-9221261135586623488/121324931\) | \(-585612268875179\) | \([]\) | \(580608\) | \(2.3920\) | \(\Gamma_0(N)\)-optimal |
27209.j2 | 27209a2 | \([0, 1, 1, -6965729, 8630952880]\) | \(-7743965038771437568/2189290237869371\) | \(-10567285823760020767139\) | \([]\) | \(1741824\) | \(2.9413\) |
Rank
sage: E.rank()
The elliptic curves in class 27209a have rank \(1\).
Complex multiplication
The elliptic curves in class 27209a do not have complex multiplication.Modular form 27209.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.