Properties

Label 570g
Number of curves $4$
Conductor $570$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("570.g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 570g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
570.g3 570g1 [1, 1, 1, -31, 53] [4] 64 \(\Gamma_0(N)\)-optimal
570.g2 570g2 [1, 1, 1, -51, -51] [2, 2] 128  
570.g1 570g3 [1, 1, 1, -621, -6207] [2] 256  
570.g4 570g4 [1, 1, 1, 199, -151] [2] 256  

Rank

sage: E.rank()
 

The elliptic curves in class 570g have rank \(0\).

Modular form 570.2.a.g

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} + 4q^{11} - q^{12} + 2q^{13} + q^{15} + q^{16} + 2q^{17} + q^{18} - q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.