Properties

Label 1710h
Number of curves $4$
Conductor $1710$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("1710.i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1710h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1710.i3 1710h1 [1, -1, 0, -279, -1715] [2] 512 \(\Gamma_0(N)\)-optimal
1710.i2 1710h2 [1, -1, 0, -459, 913] [2, 2] 1024  
1710.i1 1710h3 [1, -1, 0, -5589, 161995] [2] 2048  
1710.i4 1710h4 [1, -1, 0, 1791, 5863] [2] 2048  

Rank

sage: E.rank()
 

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

Modular form 1710.2.a.i

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - 4q^{11} + 2q^{13} + q^{16} - 2q^{17} - q^{19} + O(q^{20}) \)

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.