Properties

Label 12705.g
Number of curves 4
Conductor 12705
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("12705.g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12705.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12705.g1 12705n3 [1, 0, 0, -13615, 610292] [2] 23040  
12705.g2 12705n2 [1, 0, 0, -910, 8075] [2, 2] 11520  
12705.g3 12705n1 [1, 0, 0, -305, -1968] [2] 5760 \(\Gamma_0(N)\)-optimal
12705.g4 12705n4 [1, 0, 0, 2115, 51030] [2] 23040  

Rank

sage: E.rank()
 

The elliptic curves in class 12705.g have rank \(0\).

Modular form 12705.2.a.g

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.