Properties

Label 4998bm
Number of curves $6$
Conductor $4998$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("4998.bq1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 4998bm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4998.bq5 4998bm1 [1, 0, 0, -3441957, -2455067007] [2] 184320 \(\Gamma_0(N)\)-optimal
4998.bq4 4998bm2 [1, 0, 0, -4445477, -907037055] [2, 2] 368640  
4998.bq2 4998bm3 [1, 0, 0, -42093157, 104393523905] [2, 2] 737280  
4998.bq6 4998bm4 [1, 0, 0, 17145883, -7129667007] [2] 737280  
4998.bq1 4998bm5 [1, 0, 0, -672211597, 6708160798793] [2] 1474560  
4998.bq3 4998bm6 [1, 0, 0, -14337597, 240012741177] [2] 1474560  

Rank

sage: E.rank()
 

The elliptic curves in class 4998bm have rank \(0\).

Modular form 4998.2.a.bq

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} + 2q^{13} + 2q^{15} + q^{16} - q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.