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Properties

Label 16170e
Number of curves $8$
Conductor $16170$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 16170e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
16170.b7 16170e1 \([1, 1, 0, -57453, -2433843]\) \(178272935636041/81841914000\) \(9628619340186000\) \([2]\) \(110592\) \(1.7618\) \(\Gamma_0(N)\)-optimal
16170.b5 16170e2 \([1, 1, 0, -771873, -261196767]\) \(432288716775559561/270140062500\) \(31781708213062500\) \([2, 2]\) \(221184\) \(2.1084\)  
16170.b4 16170e3 \([1, 1, 0, -2339628, 1376376912]\) \(12038605770121350841/757333463040\) \(89099524593192960\) \([2]\) \(331776\) \(2.3111\)  
16170.b2 16170e4 \([1, 1, 0, -12348123, -16706417517]\) \(1769857772964702379561/691787250\) \(81388078175250\) \([2]\) \(442368\) \(2.4549\)  
16170.b6 16170e5 \([1, 1, 0, -626343, -362514753]\) \(-230979395175477481/348191894531250\) \(-40964428199707031250\) \([2]\) \(442368\) \(2.4549\)  
16170.b3 16170e6 \([1, 1, 0, -2480748, 1200795408]\) \(14351050585434661561/3001282273281600\) \(353097858169306958400\) \([2, 2]\) \(663552\) \(2.6577\)  
16170.b1 16170e7 \([1, 1, 0, -12564948, -16089573912]\) \(1864737106103260904761/129177711985836360\) \(15197628637421661917640\) \([2]\) \(1327104\) \(3.0042\)  
16170.b8 16170e8 \([1, 1, 0, 5345532, 7256770872]\) \(143584693754978072519/276341298967965000\) \(-32511277482282114285000\) \([2]\) \(1327104\) \(3.0042\)  

Rank

sage: E.rank()
 

The elliptic curves in class 16170e have rank \(0\).

Complex multiplication

The elliptic curves in class 16170e do not have complex multiplication.

Modular form 16170.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} - q^{11} - q^{12} - 2 q^{13} + q^{15} + q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.