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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 163254.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 163254.z1 | 163254dp3 | \([1, 1, 0, -558075014, 5074194155880]\) | \(3982367508813341135547697/173267472924\) | \(836328997716819516\) | \([2]\) | \(27525120\) | \(3.3707\) | |
| 163254.z2 | 163254dp2 | \([1, 1, 0, -34881434, 79265047620]\) | \(972403716568633470577/202799587155984\) | \(978874872480787975056\) | \([2, 2]\) | \(13762560\) | \(3.0241\) | |
| 163254.z3 | 163254dp4 | \([1, 1, 0, -31072174, 97251611488]\) | \(-687350955619188924337/448933910140540188\) | \(-2166918237871550644300092\) | \([2]\) | \(27525120\) | \(3.3707\) | |
| 163254.z4 | 163254dp1 | \([1, 1, 0, -2419914, 948384468]\) | \(324686083835773297/107226405441792\) | \(517561378824090601728\) | \([2]\) | \(6881280\) | \(2.6775\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 163254.z have rank \(1\).
Complex multiplication
The elliptic curves in class 163254.z do not have complex multiplication.Modular form 163254.2.a.z
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
