Properties

Label 605.b
Number of curves $4$
Conductor $605$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 605.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
605.b1 605b3 \([1, -1, 1, -7162, -231426]\) \(22930509321/6875\) \(12179481875\) \([2]\) \(480\) \(0.91328\)  
605.b2 605b4 \([1, -1, 1, -3532, 79786]\) \(2749884201/73205\) \(129687123005\) \([2]\) \(480\) \(0.91328\)  
605.b3 605b2 \([1, -1, 1, -507, -2494]\) \(8120601/3025\) \(5358972025\) \([2, 2]\) \(240\) \(0.56671\)  
605.b4 605b1 \([1, -1, 1, 98, -316]\) \(59319/55\) \(-97435855\) \([4]\) \(120\) \(0.22013\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 605.b have rank \(1\).

Complex multiplication

The elliptic curves in class 605.b do not have complex multiplication.

Modular form 605.2.a.b

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.