Properties

Label 2178c
Number of curves 4
Conductor 2178
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("2178.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2178c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2178.b3 2178c1 [1, -1, 0, -6012, 170748] [2] 3840 \(\Gamma_0(N)\)-optimal
2178.b4 2178c2 [1, -1, 0, 4878, 713070] [2] 7680  
2178.b1 2178c3 [1, -1, 0, -87687, -9934083] [2] 11520  
2178.b2 2178c4 [1, -1, 0, -44127, -19839627] [2] 23040  

Rank

sage: E.rank()
 

The elliptic curves in class 2178c have rank \(1\).

Modular form 2178.2.a.b

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

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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.