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Properties

Label 478584.bg
Number of curves $4$
Conductor $478584$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bg1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 478584.bg have rank \(1\).

Complex multiplication

The elliptic curves in class 478584.bg do not have complex multiplication.

Modular form 478584.2.a.bg

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{5} + 4 q^{7} - 4 q^{11} - 2 q^{13} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 478584.bg

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
478584.bg1 478584bg4 \([0, 0, 0, -1956819, -891581762]\) \(45989074372/7555707\) \(136143252389919034368\) \([2]\) \(15728640\) \(2.5850\)  
478584.bg2 478584bg2 \([0, 0, 0, -552279, 144687850]\) \(4135597648/385641\) \(1737177606109552896\) \([2, 2]\) \(7864320\) \(2.2384\)  
478584.bg3 478584bg1 \([0, 0, 0, -539274, 152425825]\) \(61604313088/621\) \(174836715590736\) \([2]\) \(3932160\) \(1.8918\) \(\Gamma_0(N)\)-optimal*
478584.bg4 478584bg3 \([0, 0, 0, 644181, 685727062]\) \(1640689628/12223143\) \(-220243908670237228032\) \([2]\) \(15728640\) \(2.5850\)  
*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 478584.bg1.