Properties

Label 4641b
Number of curves $6$
Conductor $4641$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4641b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4641.b4 4641b1 \([1, 1, 1, -1644, 24972]\) \(491411892194497/78897\) \(78897\) \([4]\) \(1536\) \(0.34217\) \(\Gamma_0(N)\)-optimal
4641.b3 4641b2 \([1, 1, 1, -1649, 24806]\) \(495909170514577/6224736609\) \(6224736609\) \([2, 4]\) \(3072\) \(0.68874\)  
4641.b2 4641b3 \([1, 1, 1, -3094, -27214]\) \(3275619238041697/1605271262049\) \(1605271262049\) \([2, 2]\) \(6144\) \(1.0353\)  
4641.b5 4641b4 \([1, 1, 1, -284, 66302]\) \(-2533811507137/1904381781393\) \(-1904381781393\) \([4]\) \(6144\) \(1.0353\)  
4641.b1 4641b5 \([1, 1, 1, -40579, -3160960]\) \(7389727131216686257/6115533215337\) \(6115533215337\) \([2]\) \(12288\) \(1.3819\)  
4641.b6 4641b6 \([1, 1, 1, 11271, -193848]\) \(158346567380527343/108665074944153\) \(-108665074944153\) \([2]\) \(12288\) \(1.3819\)  

Rank

sage: E.rank()
 

The elliptic curves in class 4641b have rank \(1\).

Complex multiplication

The elliptic curves in class 4641b do not have complex multiplication.

Modular form 4641.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} - q^{7} + 3q^{8} + q^{9} + 2q^{10} - 4q^{11} + q^{12} + q^{13} + q^{14} + 2q^{15} - q^{16} + q^{17} - q^{18} - 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.