Properties

Label 585.g
Number of curves $8$
Conductor $585$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 585.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
585.g1 585f7 [1, -1, 0, -1170000, 487402461] [2] 3072  
585.g2 585f5 [1, -1, 0, -73125, 7629336] [2, 2] 1536  
585.g3 585f8 [1, -1, 0, -71370, 8011575] [2] 3072  
585.g4 585f3 [1, -1, 0, -4680, 114075] [2, 2] 768  
585.g5 585f2 [1, -1, 0, -1035, -10584] [2, 2] 384  
585.g6 585f1 [1, -1, 0, -990, -11745] [2] 192 \(\Gamma_0(N)\)-optimal
585.g7 585f4 [1, -1, 0, 1890, -61479] [2] 768  
585.g8 585f6 [1, -1, 0, 5445, 533250] [2] 1536  

Rank

sage: E.rank()
 

The elliptic curves in class 585.g have rank \(1\).

Complex multiplication

The elliptic curves in class 585.g do not have complex multiplication.

Modular form 585.2.a.g

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.