Properties

Label 27456bz
Number of curves $6$
Conductor $27456$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("27456.cf1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 27456bz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
27456.cf5 27456bz1 [0, 1, 0, -1537, -33793] [2] 32768 \(\Gamma_0(N)\)-optimal
27456.cf4 27456bz2 [0, 1, 0, -27457, -1760065] [2, 2] 65536  
27456.cf3 27456bz3 [0, 1, 0, -30337, -1371265] [2, 2] 131072  
27456.cf1 27456bz4 [0, 1, 0, -439297, -112215553] [2] 131072  
27456.cf6 27456bz5 [0, 1, 0, 85823, -9246913] [2] 262144  
27456.cf2 27456bz6 [0, 1, 0, -192577, 31433663] [2] 262144  

Rank

sage: E.rank()
 

The elliptic curves in class 27456bz have rank \(1\).

Modular form 27456.2.a.cf

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

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 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.