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Properties

Label 164730q
Number of curves $4$
Conductor $164730$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("q1")
 
E.isogeny_class()
 

Elliptic curves in class 164730q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
164730.cr3 164730q1 \([1, 1, 1, -7476725, 7938046235]\) \(-1914980734749238129/20440940544000\) \(-493394612805697536000\) \([2]\) \(13271040\) \(2.7871\) \(\Gamma_0(N)\)-optimal
164730.cr2 164730q2 \([1, 1, 1, -119932405, 505486956827]\) \(7903870428425797297009/886464000000\) \(21397085966016000000\) \([2]\) \(26542080\) \(3.1337\)  
164730.cr4 164730q3 \([1, 1, 1, 24706315, 41347592987]\) \(69096190760262356111/70568821500000000\) \(-1703359798204933500000000\) \([2]\) \(39813120\) \(3.3364\)  
164730.cr1 164730q4 \([1, 1, 1, -133873765, 380645532155]\) \(10993009831928446009969/3767761230468750000\) \(90944596675964355468750000\) \([2]\) \(79626240\) \(3.6830\)  

Rank

sage: E.rank()
 

The elliptic curves in class 164730q have rank \(2\).

Complex multiplication

The elliptic curves in class 164730q do not have complex multiplication.

Modular form 164730.2.a.q

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2 q^{7} + q^{8} + q^{9} + q^{10} - 6 q^{11} - q^{12} - 4 q^{13} - 2 q^{14} - q^{15} + q^{16} + q^{18} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.