Properties

Label 35904.cf
Number of curves $4$
Conductor $35904$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cf1")
 
E.isogeny_class()
 

Elliptic curves in class 35904.cf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
35904.cf1 35904bg4 \([0, 1, 0, -123969, -15612129]\) \(803760366578833/65593817586\) \(17195025717264384\) \([2]\) \(294912\) \(1.8580\)  
35904.cf2 35904bg2 \([0, 1, 0, -26049, 1328031]\) \(7457162887153/1370924676\) \(359379678265344\) \([2, 2]\) \(147456\) \(1.5114\)  
35904.cf3 35904bg1 \([0, 1, 0, -24769, 1492127]\) \(6411014266033/296208\) \(77649149952\) \([2]\) \(73728\) \(1.1648\) \(\Gamma_0(N)\)-optimal
35904.cf4 35904bg3 \([0, 1, 0, 51391, 7786527]\) \(57258048889007/132611470002\) \(-34763301192204288\) \([4]\) \(294912\) \(1.8580\)  

Rank

sage: E.rank()
 

The elliptic curves in class 35904.cf have rank \(1\).

Complex multiplication

The elliptic curves in class 35904.cf do not have complex multiplication.

Modular form 35904.2.a.cf

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} + 4 q^{7} + q^{9} - q^{11} + 2 q^{13} - 2 q^{15} + q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.