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SageMath
E = EllipticCurve("fj1")
E.isogeny_class()
Elliptic curves in class 84150fj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 84150.dw3 | 84150fj1 | \([1, -1, 1, -87080, -9868453]\) | \(6411014266033/296208\) | \(3373994250000\) | \([2]\) | \(393216\) | \(1.4791\) | \(\Gamma_0(N)\)-optimal |
| 84150.dw2 | 84150fj2 | \([1, -1, 1, -91580, -8788453]\) | \(7457162887153/1370924676\) | \(15615688887562500\) | \([2, 2]\) | \(786432\) | \(1.8257\) | |
| 84150.dw4 | 84150fj3 | \([1, -1, 1, 180670, -51259453]\) | \(57258048889007/132611470002\) | \(-1510527525491531250\) | \([2]\) | \(1572864\) | \(2.1723\) | |
| 84150.dw1 | 84150fj4 | \([1, -1, 1, -435830, 102748547]\) | \(803760366578833/65593817586\) | \(747154578440531250\) | \([2]\) | \(1572864\) | \(2.1723\) |
Rank
sage: E.rank()
The elliptic curves in class 84150fj have rank \(0\).
Complex multiplication
The elliptic curves in class 84150fj do not have complex multiplication.Modular form 84150.2.a.fj
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.
