Properties

Label 111090r
Number of curves 8
Conductor 111090
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("111090.u1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 111090r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
111090.u7 111090r1 [1, 0, 1, -263189, -51971488] [2] 1216512 \(\Gamma_0(N)\)-optimal
111090.u6 111090r2 [1, 0, 1, -305509, -34146304] [2, 2] 2433024  
111090.u5 111090r3 [1, 0, 1, -778964, 201043922] [2] 3649536  
111090.u8 111090r4 [1, 0, 1, 1016991, -250507304] [2] 4866048  
111090.u4 111090r5 [1, 0, 1, -2305129, 1323195752] [2] 4866048  
111090.u2 111090r6 [1, 0, 1, -11612884, 15229857746] [2, 2] 7299072  
111090.u3 111090r7 [1, 0, 1, -10766484, 17544253906] [2] 14598144  
111090.u1 111090r8 [1, 0, 1, -185802004, 974802882002] [2] 14598144  

Rank

sage: E.rank()
 

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

Modular form 111090.2.a.u

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.