Properties

Label 111090.di
Number of curves $8$
Conductor $111090$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("111090.di1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 111090.di

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
111090.di1 111090dd8 [1, 0, 0, -3412590, -1526860650] [2] 7299072  
111090.di2 111090dd5 [1, 0, 0, -3047580, -2048021928] [2] 2433024  
111090.di3 111090dd6 [1, 0, 0, -1428840, 639791100] [2, 2] 3649536  
111090.di4 111090dd3 [1, 0, 0, -1418260, 649983872] [2] 1824768  
111090.di5 111090dd2 [1, 0, 0, -190980, -31833648] [2, 2] 1216512  
111090.di6 111090dd4 [1, 0, 0, -42860, -79913400] [2] 2433024  
111090.di7 111090dd1 [1, 0, 0, -21700, 431120] [2] 608256 \(\Gamma_0(N)\)-optimal
111090.di8 111090dd7 [1, 0, 0, 385630, 2154147762] [2] 7299072  

Rank

sage: E.rank()
 

The elliptic curves in class 111090.di have rank \(1\).

Modular form 111090.2.a.di

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

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.