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Properties

Label 455175.bu
Number of curves $6$
Conductor $455175$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bu1")
 
E.isogeny_class()
 

Elliptic curves in class 455175.bu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
455175.bu1 455175bu5 \([1, -1, 1, -4664894855, 122397359564522]\) \(40832710302042509761/91556816413125\) \(25172813933572937715439453125\) \([2]\) \(452984832\) \(4.3318\) \(\Gamma_0(N)\)-optimal*
455175.bu2 455175bu3 \([1, -1, 1, -397629230, 396235345772]\) \(25288177725059761/14387797265625\) \(3955809711068411627197265625\) \([2, 2]\) \(226492416\) \(3.9853\) \(\Gamma_0(N)\)-optimal*
455175.bu3 455175bu2 \([1, -1, 1, -254249105, -1553160833728]\) \(6610905152742241/35128130625\) \(9658198381072218837890625\) \([2, 2]\) \(113246208\) \(3.6387\) \(\Gamma_0(N)\)-optimal*
455175.bu4 455175bu1 \([1, -1, 1, -253923980, -1557349093978]\) \(6585576176607121/187425\) \(51531003767225390625\) \([2]\) \(56623104\) \(3.2921\) \(\Gamma_0(N)\)-optimal*
455175.bu5 455175bu4 \([1, -1, 1, -116070980, -3234512258728]\) \(-629004249876241/16074715228425\) \(-4419614304351308686616015625\) \([2]\) \(226492416\) \(3.9853\)  
455175.bu6 455175bu6 \([1, -1, 1, 1575554395, 3154746053522]\) \(1573196002879828319/926055908203125\) \(-254611660633728504180908203125\) \([2]\) \(452984832\) \(4.3318\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 455175.bu1.

Rank

sage: E.rank()
 

The elliptic curves in class 455175.bu have rank \(1\).

Complex multiplication

The elliptic curves in class 455175.bu do not have complex multiplication.

Modular form 455175.2.a.bu

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + q^{7} + 3 q^{8} - 4 q^{11} + 2 q^{13} - q^{14} - q^{16} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.