Properties

Label 690f
Number of curves $4$
Conductor $690$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 690f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.f3 690f1 \([1, 0, 1, -13, 8]\) \(217081801/88320\) \(88320\) \([2]\) \(64\) \(-0.35322\) \(\Gamma_0(N)\)-optimal
690.f2 690f2 \([1, 0, 1, -93, -344]\) \(87587538121/1904400\) \(1904400\) \([2, 2]\) \(128\) \(-0.0066500\)  
690.f1 690f3 \([1, 0, 1, -1473, -21872]\) \(353108405631241/172500\) \(172500\) \([2]\) \(256\) \(0.33992\)  
690.f4 690f4 \([1, 0, 1, 7, -1024]\) \(46268279/453342420\) \(-453342420\) \([4]\) \(256\) \(0.33992\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 690.2.a.f

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.