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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 439824.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
439824.r1 | 439824r4 | \([0, -1, 0, -1518624, -667092096]\) | \(803760366578833/65593817586\) | \(31609024697038086144\) | \([2]\) | \(10616832\) | \(2.4844\) | |
439824.r2 | 439824r2 | \([0, -1, 0, -319104, 57417984]\) | \(7457162887153/1370924676\) | \(660635308878741504\) | \([2, 2]\) | \(5308416\) | \(2.1378\) | |
439824.r3 | 439824r1 | \([0, -1, 0, -303424, 64430080]\) | \(6411014266033/296208\) | \(142739763167232\) | \([2]\) | \(2654208\) | \(1.7912\) | \(\Gamma_0(N)\)-optimal* |
439824.r4 | 439824r3 | \([0, -1, 0, 629536, 332903040]\) | \(57258048889007/132611470002\) | \(-63904181593150660608\) | \([2]\) | \(10616832\) | \(2.4844\) |
Rank
sage: E.rank()
The elliptic curves in class 439824.r have rank \(1\).
Complex multiplication
The elliptic curves in class 439824.r do not have complex multiplication.Modular form 439824.2.a.r
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.