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SageMath
E = EllipticCurve("ka1")
E.isogeny_class()
Elliptic curves in class 177450ka
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
177450.bc2 | 177450ka1 | \([1, 1, 0, -5649891900, -163556553438000]\) | \(-44694151057272491356949809/30197762286189281280\) | \(-13476223260247688478720000000\) | \([]\) | \(239500800\) | \(4.3357\) | \(\Gamma_0(N)\)-optimal |
177450.bc1 | 177450ka2 | \([1, 1, 0, -457715427900, -119190635459550000]\) | \(-23763856998804796987128199384369/7318708992000\) | \(-3266088242508000000000\) | \([]\) | \(718502400\) | \(4.8850\) |
Rank
sage: E.rank()
The elliptic curves in class 177450ka have rank \(1\).
Complex multiplication
The elliptic curves in class 177450ka do not have complex multiplication.Modular form 177450.2.a.ka
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.