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Properties

Label 3570a
Number of curves $2$
Conductor $3570$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 3570a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3570.b2 3570a1 \([1, 1, 0, -1683, -70227]\) \(-527690404915129/1782829440000\) \(-1782829440000\) \([2]\) \(7680\) \(1.0369\) \(\Gamma_0(N)\)-optimal
3570.b1 3570a2 \([1, 1, 0, -37683, -2827827]\) \(5918043195362419129/8515734343200\) \(8515734343200\) \([2]\) \(15360\) \(1.3835\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 3570.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.