Properties

Label 12138.bc
Number of curves $6$
Conductor $12138$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("12138.bc1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12138.bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12138.bc1 12138bb4 [1, 0, 0, -388422, 93143592] [2] 81920  
12138.bc2 12138bb5 [1, 0, 0, -264152, -51775458] [2] 163840  
12138.bc3 12138bb3 [1, 0, 0, -30062, 707520] [2, 2] 81920  
12138.bc4 12138bb2 [1, 0, 0, -24282, 1453140] [2, 2] 40960  
12138.bc5 12138bb1 [1, 0, 0, -1162, 33572] [2] 20480 \(\Gamma_0(N)\)-optimal
12138.bc6 12138bb6 [1, 0, 0, 111548, 5493938] [2] 163840  

Rank

sage: E.rank()
 

The elliptic curves in class 12138.bc have rank \(0\).

Modular form 12138.2.a.bc

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} + 6q^{13} + q^{14} + 2q^{15} + q^{16} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.