Properties

Label 30912bk
Number of curves $6$
Conductor $30912$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("30912.w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 30912bk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
30912.w5 30912bk1 [0, -1, 0, 8063, -573407] [2] 98304 \(\Gamma_0(N)\)-optimal
30912.w4 30912bk2 [0, -1, 0, -73857, -6586335] [2, 2] 196608  
30912.w3 30912bk3 [0, -1, 0, -324737, 64914465] [2, 2] 393216  
30912.w2 30912bk4 [0, -1, 0, -1133697, -464225247] [2] 393216  
30912.w6 30912bk5 [0, -1, 0, 401023, 313269537] [2] 786432  
30912.w1 30912bk6 [0, -1, 0, -5064577, 4388596513] [2] 786432  

Rank

sage: E.rank()
 

The elliptic curves in class 30912bk have rank \(1\).

Modular form 30912.2.a.w

sage: E.q_eigenform(10)
 
\( q - q^{3} + 2q^{5} - q^{7} + q^{9} - 4q^{11} + 2q^{13} - 2q^{15} - 6q^{17} + 4q^{19} + O(q^{20}) \)

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.