Properties

Label 324870el
Number of curves 8
Conductor 324870
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("324870.el1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 324870el

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
324870.el7 324870el1 [1, 1, 1, -102711155, -385844310223] [2] 79626240 \(\Gamma_0(N)\)-optimal
324870.el6 324870el2 [1, 1, 1, -272306035, 1216420278065] [2, 2] 159252480  
324870.el5 324870el3 [1, 1, 1, -1263846515, 17180605977905] [2] 238878720  
324870.el4 324870el4 [1, 1, 1, -3987290035, 96895090201265] [2] 318504960  
324870.el8 324870el5 [1, 1, 1, 729159885, 8086075902897] [2] 318504960  
324870.el2 324870el6 [1, 1, 1, -20184937795, 1103788604442257] [2, 2] 477757440  
324870.el1 324870el7 [1, 1, 1, -322959000295, 70642876415817257] [2] 955514880  
324870.el3 324870el8 [1, 1, 1, -20148335775, 1107991116611385] [2] 955514880  

Rank

sage: E.rank()
 

The elliptic curves in class 324870el have rank \(0\).

Modular form 324870.2.a.el

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.