Properties

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

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Show commands: 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 j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
585.g1 585f7 \([1, -1, 0, -1170000, 487402461]\) \(242970740812818720001/24375\) \(17769375\) \([2]\) \(3072\) \(1.7387\)  
585.g2 585f5 \([1, -1, 0, -73125, 7629336]\) \(59319456301170001/594140625\) \(433128515625\) \([2, 2]\) \(1536\) \(1.3921\)  
585.g3 585f8 \([1, -1, 0, -71370, 8011575]\) \(-55150149867714721/5950927734375\) \(-4338226318359375\) \([2]\) \(3072\) \(1.7387\)  
585.g4 585f3 \([1, -1, 0, -4680, 114075]\) \(15551989015681/1445900625\) \(1054061555625\) \([2, 2]\) \(768\) \(1.0456\)  
585.g5 585f2 \([1, -1, 0, -1035, -10584]\) \(168288035761/27720225\) \(20208044025\) \([2, 2]\) \(384\) \(0.69900\)  
585.g6 585f1 \([1, -1, 0, -990, -11745]\) \(147281603041/5265\) \(3838185\) \([2]\) \(192\) \(0.35243\) \(\Gamma_0(N)\)-optimal
585.g7 585f4 \([1, -1, 0, 1890, -61479]\) \(1023887723039/2798036865\) \(-2039768874585\) \([2]\) \(768\) \(1.0456\)  
585.g8 585f6 \([1, -1, 0, 5445, 533250]\) \(24487529386319/183539412225\) \(-133800231512025\) \([2]\) \(1536\) \(1.3921\)  

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)
 
\(q + q^{2} - q^{4} - q^{5} - 3 q^{8} - q^{10} - 4 q^{11} + q^{13} - q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.