Properties

Label 690a
Number of curves $2$
Conductor $690$
CM no
Rank $1$
Graph

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

Elliptic curves in class 690a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.a2 690a1 \([1, 1, 0, 172, -1968]\) \(557644990391/2119680000\) \(-2119680000\) \([2]\) \(448\) \(0.47234\) \(\Gamma_0(N)\)-optimal
690.a1 690a2 \([1, 1, 0, -1748, -25392]\) \(591202341974089/79350000000\) \(79350000000\) \([2]\) \(896\) \(0.81892\)  

Rank

sage: E.rank()
 

The elliptic curves in class 690a have rank \(1\).

Complex multiplication

The elliptic curves in class 690a do not have complex multiplication.

Modular form 690.2.a.a

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