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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 292142.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
292142.f1 | 292142f3 | \([1, 1, 1, -9645347, 11525850945]\) | \(15698803397448457/20709376\) | \(130911484214247424\) | \([]\) | \(9434880\) | \(2.5605\) | |
292142.f2 | 292142f2 | \([1, 1, 1, -150732, 6691885]\) | \(59914169497/31554496\) | \(199467425044218304\) | \([]\) | \(3144960\) | \(2.0112\) | |
292142.f3 | 292142f1 | \([1, 1, 1, -86017, -9745725]\) | \(11134383337/316\) | \(1997550723484\) | \([]\) | \(1048320\) | \(1.4618\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 292142.f have rank \(2\).
Complex multiplication
The elliptic curves in class 292142.f do not have complex multiplication.Modular form 292142.2.a.f
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.