Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 570.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
570.f1 570f4 \([1, 0, 1, -7478, -240952]\) \(46237740924063961/1806561830400\) \(1806561830400\) \([2]\) \(864\) \(1.1191\)  
570.f2 570f2 \([1, 0, 1, -1103, 13898]\) \(148212258825961/1218375000\) \(1218375000\) \([6]\) \(288\) \(0.56979\)  
570.f3 570f1 \([1, 0, 1, -23, 506]\) \(-1263214441/110808000\) \(-110808000\) \([6]\) \(144\) \(0.22322\) \(\Gamma_0(N)\)-optimal
570.f4 570f3 \([1, 0, 1, 202, -13624]\) \(918046641959/80912056320\) \(-80912056320\) \([2]\) \(432\) \(0.77253\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 570.f do not have complex multiplication.

Modular form 570.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 2q^{7} - q^{8} + q^{9} - q^{10} + q^{12} + 2q^{13} - 2q^{14} + q^{15} + q^{16} - q^{18} + q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.