Properties

Label 570.e
Number of curves $4$
Conductor $570$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 570.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
570.e1 570e3 \([1, 0, 1, -968, -11662]\) \(100162392144121/23457780\) \(23457780\) \([2]\) \(512\) \(0.40469\)  
570.e2 570e4 \([1, 0, 1, -448, 3506]\) \(9912050027641/311647500\) \(311647500\) \([4]\) \(512\) \(0.40469\)  
570.e3 570e2 \([1, 0, 1, -68, -142]\) \(34043726521/11696400\) \(11696400\) \([2, 2]\) \(256\) \(0.058114\)  
570.e4 570e1 \([1, 0, 1, 12, -14]\) \(214921799/218880\) \(-218880\) \([2]\) \(128\) \(-0.28846\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 570.e have rank \(1\).

Complex multiplication

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

Modular form 570.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 4q^{7} - q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} - 6q^{13} + 4q^{14} + q^{15} + q^{16} - 6q^{17} - q^{18} - q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.