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SageMath
E = EllipticCurve("hk1")
E.isogeny_class()
Elliptic curves in class 43200.hk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43200.hk1 | 43200ih1 | \([0, 0, 0, -254700, -49486000]\) | \(-16522921323/4000\) | \(-442368000000000\) | \([]\) | \(276480\) | \(1.7991\) | \(\Gamma_0(N)\)-optimal |
43200.hk2 | 43200ih2 | \([0, 0, 0, 105300, -174366000]\) | \(1601613/163840\) | \(-13209037701120000000\) | \([]\) | \(829440\) | \(2.3484\) |
Rank
sage: E.rank()
The elliptic curves in class 43200.hk have rank \(1\).
Complex multiplication
The elliptic curves in class 43200.hk do not have complex multiplication.Modular form 43200.2.a.hk
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.