Properties

Label 76230.ea
Number of curves $8$
Conductor $76230$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("76230.ea1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 76230.ea

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
76230.ea1 76230ek8 [1, -1, 1, -7025162, 4510720811] [2] 6635520  
76230.ea2 76230ek5 [1, -1, 1, -6273752, 6049948259] [2] 2211840  
76230.ea3 76230ek6 [1, -1, 1, -2941412, -1889332189] [2, 2] 3317760  
76230.ea4 76230ek3 [1, -1, 1, -2919632, -1919440861] [2] 1658880  
76230.ea5 76230ek2 [1, -1, 1, -393152, 94076579] [2, 2] 1105920  
76230.ea6 76230ek4 [1, -1, 1, -88232, 236047331] [2] 2211840  
76230.ea7 76230ek1 [1, -1, 1, -44672, -1267549] [2] 552960 \(\Gamma_0(N)\)-optimal
76230.ea8 76230ek7 [1, -1, 1, 793858, -6362691541] [2] 6635520  

Rank

sage: E.rank()
 

The elliptic curves in class 76230.ea have rank \(1\).

Modular form 76230.2.a.ea

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

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.