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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 179520dz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 179520.x4 | 179520dz1 | \([0, -1, 0, -227041, 6906241]\) | \(4937402992298041/2780405760000\) | \(728866687549440000\) | \([2]\) | \(1769472\) | \(2.1176\) | \(\Gamma_0(N)\)-optimal |
| 179520.x2 | 179520dz2 | \([0, -1, 0, -2275041, -1313234559]\) | \(4967657717692586041/29490113030400\) | \(7730656190241177600\) | \([2, 2]\) | \(3538944\) | \(2.4642\) | |
| 179520.x3 | 179520dz3 | \([0, -1, 0, -969441, -2811279999]\) | \(-384369029857072441/12804787777021680\) | \(-3356698287019571281920\) | \([2]\) | \(7077888\) | \(2.8108\) | |
| 179520.x1 | 179520dz4 | \([0, -1, 0, -36348641, -84336968319]\) | \(20260414982443110947641/720358602480\) | \(188837685488517120\) | \([2]\) | \(7077888\) | \(2.8108\) |
Rank
sage: E.rank()
The elliptic curves in class 179520dz have rank \(2\).
Complex multiplication
The elliptic curves in class 179520dz do not have complex multiplication.Modular form 179520.2.a.dz
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.
