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Properties

Label 118354e
Number of curves $4$
Conductor $118354$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 118354e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
118354.b4 118354e1 [1, 0, 1, -10516, -257838] [2] 417600 \(\Gamma_0(N)\)-optimal
118354.b3 118354e2 [1, 0, 1, -149756, -22313454] [2] 835200  
118354.b2 118354e3 [1, 0, 1, -358616, 82617810] [2] 1252800  
118354.b1 118354e4 [1, 0, 1, -393426, 65602682] [2] 2505600  

Rank

sage: E.rank()
 

The elliptic curves in class 118354e have rank \(2\).

Complex multiplication

The elliptic curves in class 118354e do not have complex multiplication.

Modular form 118354.2.a.e

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

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.