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SageMath
E = EllipticCurve("ix1")
E.isogeny_class()
Elliptic curves in class 430950ix
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
430950.ix4 | 430950ix1 | \([1, 0, 0, 6675412, 1126203792]\) | \(436192097814719/259683840000\) | \(-19585067126040000000000\) | \([2]\) | \(46448640\) | \(2.9667\) | \(\Gamma_0(N)\)-optimal* |
430950.ix3 | 430950ix2 | \([1, 0, 0, -27124588, 9069203792]\) | \(29263955267177281/16463793153600\) | \(1241681015123982225000000\) | \([2, 2]\) | \(92897280\) | \(3.3133\) | \(\Gamma_0(N)\)-optimal* |
430950.ix1 | 430950ix3 | \([1, 0, 0, -323719588, 2237980628792]\) | \(49745123032831462081/97939634471640\) | \(7386498580069096824375000\) | \([2]\) | \(185794560\) | \(3.6599\) | \(\Gamma_0(N)\)-optimal* |
430950.ix2 | 430950ix4 | \([1, 0, 0, -271329588, -1711355021208]\) | \(29291056630578924481/175463302795560\) | \(13233247642239596375625000\) | \([2]\) | \(185794560\) | \(3.6599\) |
Rank
sage: E.rank()
The elliptic curves in class 430950ix have rank \(0\).
Complex multiplication
The elliptic curves in class 430950ix do not have complex multiplication.Modular form 430950.2.a.ix
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.