Properties

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

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("l1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1710.l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1710.l1 1710n4 \([1, -1, 1, -8708, 314867]\) \(100162392144121/23457780\) \(17100721620\) \([2]\) \(4096\) \(0.95399\)  
1710.l2 1710n3 \([1, -1, 1, -4028, -94669]\) \(9912050027641/311647500\) \(227191027500\) \([2]\) \(4096\) \(0.95399\)  
1710.l3 1710n2 \([1, -1, 1, -608, 3827]\) \(34043726521/11696400\) \(8526675600\) \([2, 2]\) \(2048\) \(0.60742\)  
1710.l4 1710n1 \([1, -1, 1, 112, 371]\) \(214921799/218880\) \(-159563520\) \([2]\) \(1024\) \(0.26085\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1710.2.a.l

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.