Properties

Label 31824.bg
Number of curves $6$
Conductor $31824$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("31824.bg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 31824.bg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
31824.bg1 31824bk6 [0, 0, 0, -2905059, -1905520862] [2] 524288  
31824.bg2 31824bk4 [0, 0, 0, -200019, -23354030] [2, 2] 262144  
31824.bg3 31824bk2 [0, 0, 0, -78339, 8161090] [2, 2] 131072  
31824.bg4 31824bk1 [0, 0, 0, -77619, 8323378] [2] 65536 \(\Gamma_0(N)\)-optimal
31824.bg5 31824bk3 [0, 0, 0, 31821, 29289778] [2] 262144  
31824.bg6 31824bk5 [0, 0, 0, 558141, -158154878] [2] 524288  

Rank

sage: E.rank()
 

The elliptic curves in class 31824.bg have rank \(0\).

Modular form 31824.2.a.bg

sage: E.q_eigenform(10)
 
\( q + 2q^{5} + 4q^{11} + q^{13} - q^{17} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.