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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 450840.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450840.x1 | 450840x4 | \([0, -1, 0, -18061440, -29538245700]\) | \(26362547147244676/244298925\) | \(6038304930624844800\) | \([2]\) | \(21233664\) | \(2.7674\) | |
450840.x2 | 450840x2 | \([0, -1, 0, -1154940, -438777900]\) | \(27572037674704/2472575625\) | \(15278582977575840000\) | \([2, 2]\) | \(10616832\) | \(2.4208\) | |
450840.x3 | 450840x1 | \([0, -1, 0, -251815, 40962100]\) | \(4572531595264/776953125\) | \(300060154631250000\) | \([4]\) | \(5308416\) | \(2.0742\) | \(\Gamma_0(N)\)-optimal* |
450840.x4 | 450840x3 | \([0, -1, 0, 1301560, -2057120100]\) | \(9865576607324/79640206425\) | \(-1968456681223865164800\) | \([2]\) | \(21233664\) | \(2.7674\) |
Rank
sage: E.rank()
The elliptic curves in class 450840.x have rank \(1\).
Complex multiplication
The elliptic curves in class 450840.x do not have complex multiplication.Modular form 450840.2.a.x
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.