Properties

Label 1785.c
Number of curves 4
Conductor 1785
CM no
Rank 1
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("1785.c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1785.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1785.c1 1785k4 [1, 0, 0, -1686, 26505] [2] 1024  
1785.c2 1785k3 [1, 0, 0, -536, -4485] [2] 1024  
1785.c3 1785k2 [1, 0, 0, -111, 360] [2, 2] 512  
1785.c4 1785k1 [1, 0, 0, 14, 35] [2] 256 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1785.c have rank \(1\).

Modular form 1785.2.a.c

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + q^{7} + 3q^{8} + q^{9} + q^{10} - q^{12} - 6q^{13} - q^{14} - q^{15} - q^{16} - q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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 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.