Properties

Label 1827.d
Number of curves $6$
Conductor $1827$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("1827.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1827.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1827.d1 1827b3 [1, -1, 0, -1841616, -961476971] [2] 12288  
1827.d2 1827b5 [1, -1, 0, -382221, 74033554] [2] 24576  
1827.d3 1827b4 [1, -1, 0, -117306, -14395073] [2, 2] 12288  
1827.d4 1827b2 [1, -1, 0, -115101, -15001448] [2, 2] 6144  
1827.d5 1827b1 [1, -1, 0, -7056, -242501] [2] 3072 \(\Gamma_0(N)\)-optimal
1827.d6 1827b6 [1, -1, 0, 112329, -64042160] [2] 24576  

Rank

sage: E.rank()
 

The elliptic curves in class 1827.d have rank \(1\).

Modular form 1827.2.a.d

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{4} + 2q^{5} + q^{7} - 3q^{8} + 2q^{10} - 4q^{11} - 2q^{13} + q^{14} - q^{16} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.