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SageMath
E = EllipticCurve("fz1")
E.isogeny_class()
Elliptic curves in class 76050.fz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76050.fz1 | 76050ey4 | \([1, -1, 1, -788620280, -8523931241653]\) | \(986551739719628473/111045168\) | \(6105304563893700750000\) | \([2]\) | \(27525120\) | \(3.6046\) | |
| 76050.fz2 | 76050ey3 | \([1, -1, 1, -88960280, 109556790347]\) | \(1416134368422073/725251155408\) | \(39874577785155334523250000\) | \([2]\) | \(27525120\) | \(3.6046\) | |
| 76050.fz3 | 76050ey2 | \([1, -1, 1, -49414280, -132464729653]\) | \(242702053576633/2554695936\) | \(140458131032063364000000\) | \([2, 2]\) | \(13762560\) | \(3.2580\) | |
| 76050.fz4 | 76050ey1 | \([1, -1, 1, -742280, -5138777653]\) | \(-822656953/207028224\) | \(-11382488617982976000000\) | \([2]\) | \(6881280\) | \(2.9114\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 76050.fz have rank \(1\).
Complex multiplication
The elliptic curves in class 76050.fz do not have complex multiplication.Modular form 76050.2.a.fz
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
