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Properties

Label 485520.b
Number of curves $6$
Conductor $485520$
CM no
Rank $1$
Graph

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Copy content sage:E = EllipticCurve("b1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 485520.b have rank \(1\).

Complex multiplication

The elliptic curves in class 485520.b do not have complex multiplication.

Modular form 485520.2.a.b

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - q^{7} + q^{9} - 4 q^{11} - 2 q^{13} + q^{15} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 485520.b

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485520.b1 485520b5 \([0, -1, 0, -3028816, 2029849696]\) \(62161150998242/1607445\) \(79462020303267840\) \([2]\) \(10485760\) \(2.3495\) \(\Gamma_0(N)\)-optimal*
485520.b2 485520b3 \([0, -1, 0, -196616, 29183616]\) \(34008619684/4862025\) \(120174043051238400\) \([2, 2]\) \(5242880\) \(2.0029\) \(\Gamma_0(N)\)-optimal*
485520.b3 485520b2 \([0, -1, 0, -52116, -4109184]\) \(2533446736/275625\) \(1703146868640000\) \([2, 2]\) \(2621440\) \(1.6564\) \(\Gamma_0(N)\)-optimal*
485520.b4 485520b1 \([0, -1, 0, -50671, -4373330]\) \(37256083456/525\) \(202755579600\) \([2]\) \(1310720\) \(1.3098\) \(\Gamma_0(N)\)-optimal*
485520.b5 485520b4 \([0, -1, 0, 69264, -20519760]\) \(1486779836/8203125\) \(-202755579600000000\) \([2]\) \(5242880\) \(2.0029\)  
485520.b6 485520b6 \([0, -1, 0, 323584, 156944736]\) \(75798394558/259416045\) \(-12823905660712151040\) \([2]\) \(10485760\) \(2.3495\)  
*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 485520.b1.