Properties

Label 1710.a
Number of curves $2$
Conductor $1710$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 1710.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1710.a1 1710a2 \([1, -1, 0, -4875, -128539]\) \(651038076963/7220000\) \(142111260000\) \([2]\) \(3840\) \(0.95431\)  
1710.a2 1710a1 \([1, -1, 0, -555, 1925]\) \(961504803/486400\) \(9573811200\) \([2]\) \(1920\) \(0.60774\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1710.a have rank \(1\).

Complex multiplication

The elliptic curves in class 1710.a do not have complex multiplication.

Modular form 1710.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - 4 q^{7} - q^{8} + q^{10} + 6 q^{11} + 4 q^{14} + q^{16} - 4 q^{17} - q^{19} + O(q^{20})\)  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 LMFDB 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 LMFDB labels.