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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 27209.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27209.i1 | 27209b3 | \([0, 1, 1, -2506212427, -48292813035840]\) | \(-360675992659311050823073792/56219378022244619\) | \(-271360199812172527190771\) | \([]\) | \(11757312\) | \(3.9008\) | |
27209.i2 | 27209b2 | \([0, 1, 1, -26963837, -83903328245]\) | \(-449167881463536812032/369990050199923699\) | \(-1785871304215443509646491\) | \([]\) | \(3919104\) | \(3.3515\) | |
27209.i3 | 27209b1 | \([0, 1, 1, 2739603, 1870924150]\) | \(471114356703100928/585612268875179\) | \(-2826638569917133873811\) | \([]\) | \(1306368\) | \(2.8022\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 27209.i have rank \(1\).
Complex multiplication
The elliptic curves in class 27209.i do not have complex multiplication.Modular form 27209.2.a.i
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.