Properties

Label 164730.a
Number of curves $4$
Conductor $164730$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 164730.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
164730.a1 164730cx3 \([1, 1, 0, -4230070608, 105891762826512]\) \(346795165011870675497264041/121778756846679600\) \(2939443146120951265892400\) \([2]\) \(212336640\) \(4.0500\)  
164730.a2 164730cx2 \([1, 1, 0, -265568608, 1638839533312]\) \(85814444987865209552041/1585867720793760000\) \(38278991535532116769440000\) \([2, 2]\) \(106168320\) \(3.7034\)  
164730.a3 164730cx1 \([1, 1, 0, -34368608, -38886386688]\) \(186001322269702352041/80595993600000000\) \(1945391356643558400000000\) \([2]\) \(53084160\) \(3.3568\) \(\Gamma_0(N)\)-optimal
164730.a4 164730cx4 \([1, 1, 0, -266608, 4764999120112]\) \(-86826493040041/406364619140547159600\) \(-9808654033663677762599012400\) \([2]\) \(212336640\) \(4.0500\)  

Rank

sage: E.rank()
 

The elliptic curves in class 164730.a have rank \(1\).

Complex multiplication

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

Modular form 164730.2.a.a

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

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

The vertices are labelled with LMFDB labels.