Properties

Label 83391r
Number of curves 6
Conductor 83391
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("83391.p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 83391r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
83391.p4 83391r1 [1, 0, 1, -12282, -524729] [2] 138240 \(\Gamma_0(N)\)-optimal
83391.p3 83391r2 [1, 0, 1, -14087, -360835] [2, 2] 276480  
83391.p6 83391r3 [1, 0, 1, 45478, -2600479] [2] 552960  
83391.p2 83391r4 [1, 0, 1, -102532, 12375245] [2, 2] 552960  
83391.p5 83391r5 [1, 0, 1, 11183, 38347751] [2] 1105920  
83391.p1 83391r6 [1, 0, 1, -1631367, 801865639] [2] 1105920  

Rank

sage: E.rank()
 

The elliptic curves in class 83391r have rank \(2\).

Modular form 83391.2.a.p

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.