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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 481338.bb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
481338.bb1 | 481338bb3 | \([1, -1, 0, -682966917, -6869680410411]\) | \(27279667585959979515625/20847389708288\) | \(26923736045514205827072\) | \([2]\) | \(169205760\) | \(3.6123\) | |
481338.bb2 | 481338bb4 | \([1, -1, 0, -678436677, -6965312870763]\) | \(-26740407923656692603625/754628826325811008\) | \(-974578957683848874779602752\) | \([2]\) | \(338411520\) | \(3.9589\) | |
481338.bb3 | 481338bb1 | \([1, -1, 0, -10297062, -4946877468]\) | \(93493211839989625/45910522026512\) | \(59291968637309216794128\) | \([2]\) | \(56401920\) | \(3.0630\) | \(\Gamma_0(N)\)-optimal* |
481338.bb4 | 481338bb2 | \([1, -1, 0, 37553598, -37935122472]\) | \(4535182051990706375/3105662922242452\) | \(-4010864186587064410019988\) | \([2]\) | \(112803840\) | \(3.4095\) |
Rank
sage: E.rank()
The elliptic curves in class 481338.bb have rank \(0\).
Complex multiplication
The elliptic curves in class 481338.bb do not have complex multiplication.Modular form 481338.2.a.bb
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.