Properties

Label 735e
Number of curves 8
Conductor 735
CM no
Rank 1
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath

sage: E = EllipticCurve("735.c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 735e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
735.c7 735e1 [1, 0, 0, -1, -64] [2] 96 \(\Gamma_0(N)\)-optimal
735.c6 735e2 [1, 0, 0, -246, -1485] [2, 2] 192  
735.c4 735e3 [1, 0, 0, -3921, -94830] [2] 384  
735.c5 735e4 [1, 0, 0, -491, 1896] [2, 2] 384  
735.c2 735e5 [1, 0, 0, -6616, 206471] [2, 2] 768  
735.c8 735e6 [1, 0, 0, 1714, 14685] [2] 768  
735.c1 735e7 [1, 0, 0, -105841, 13244636] [2] 1536  
735.c3 735e8 [1, 0, 0, -5391, 285606] [2] 1536  

Rank

sage: E.rank()
 

The elliptic curves in class 735e have rank \(1\).

Modular form 735.2.a.c

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + 3q^{8} + q^{9} + q^{10} - 4q^{11} - q^{12} + 2q^{13} - q^{15} - q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.