Properties

Label 1617g
Number of curves 6
Conductor 1617
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1617.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1617g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1617.e4 1617g1 [1, 0, 0, -1667, -26328] [2] 960 \(\Gamma_0(N)\)-optimal
1617.e3 1617g2 [1, 0, 0, -1912, -18145] [2, 2] 1920  
1617.e2 1617g3 [1, 0, 0, -13917, 618120] [2, 2] 3840  
1617.e6 1617g4 [1, 0, 0, 6173, -129718] [2] 3840  
1617.e1 1617g5 [1, 0, 0, -221432, 40087473] [4] 7680  
1617.e5 1617g6 [1, 0, 0, 1518, 1917747] [2] 7680  

Rank

sage: E.rank()
 

The elliptic curves in class 1617g have rank \(1\).

Modular form 1617.2.a.e

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.