Properties

Label 3570s
Number of curves $4$
Conductor $3570$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("s1")
 
E.isogeny_class()
 

Elliptic curves in class 3570s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3570.r4 3570s1 \([1, 1, 1, 2269, -6847]\) \(1291859362462031/773834342400\) \(-773834342400\) \([2]\) \(6144\) \(0.97051\) \(\Gamma_0(N)\)-optimal
3570.r3 3570s2 \([1, 1, 1, -9251, -66751]\) \(87557366190249649/48960807840000\) \(48960807840000\) \([2, 2]\) \(12288\) \(1.3171\)  
3570.r1 3570s3 \([1, 1, 1, -111251, -14305951]\) \(152277495831664137649/282362258900400\) \(282362258900400\) \([2]\) \(24576\) \(1.6637\)  
3570.r2 3570s4 \([1, 1, 1, -91571, 10568993]\) \(84917632843343402929/537144431250000\) \(537144431250000\) \([2]\) \(24576\) \(1.6637\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3570s have rank \(1\).

Complex multiplication

The elliptic curves in class 3570s do not have complex multiplication.

Modular form 3570.2.a.s

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