Properties

Label 690d
Number of curves $2$
Conductor $690$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 690d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.d2 690d1 \([1, 1, 0, -12, -36]\) \(-217081801/285660\) \(-285660\) \([2]\) \(144\) \(-0.26022\) \(\Gamma_0(N)\)-optimal
690.d1 690d2 \([1, 1, 0, -242, -1554]\) \(1577505447721/838350\) \(838350\) \([2]\) \(288\) \(0.086349\)  

Rank

sage: E.rank()
 

The elliptic curves in class 690d have rank \(0\).

Complex multiplication

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

Modular form 690.2.a.d

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