Properties

Label 111600.de
Number of curves $6$
Conductor $111600$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("111600.de1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 111600.de

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
111600.de1 111600ev6 [0, 0, 0, -1107072075, -14177901599750] [2] 18874368  
111600.de2 111600ev4 [0, 0, 0, -69192075, -221529239750] [2, 2] 9437184  
111600.de3 111600ev5 [0, 0, 0, -68112075, -228779279750] [4] 18874368  
111600.de4 111600ev3 [0, 0, 0, -13320075, 14588712250] [2] 9437184  
111600.de5 111600ev2 [0, 0, 0, -4392075, -3347639750] [2, 2] 4718592  
111600.de6 111600ev1 [0, 0, 0, 215925, -218807750] [2] 2359296 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 111600.de have rank \(2\).

Modular form 111600.2.a.de

sage: E.q_eigenform(10)
 
\( q - 4q^{11} - 6q^{13} + 2q^{17} - 4q^{19} + O(q^{20}) \)

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.