Properties

Label 481338.bb
Number of curves $4$
Conductor $481338$
CM no
Rank $0$
Graph

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Show commands: 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 \([1, -1, 0, 37553598, -37935122472]\) \(4535182051990706375/3105662922242452\) \(-4010864186587064410019988\) \([2]\) \(112803840\) \(3.4095\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 481338.bb1.

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)
 
\(q - q^{2} + q^{4} + 4 q^{7} - q^{8} - q^{13} - 4 q^{14} + q^{16} + q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.