Properties

Label 8400cc
Number of curves 8
Conductor 8400
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("8400.ce1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8400cc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8400.ce7 8400cc1 [0, 1, 0, 83992, -7032012] [2] 73728 \(\Gamma_0(N)\)-optimal
8400.ce6 8400cc2 [0, 1, 0, -428008, -63352012] [2, 2] 147456  
8400.ce4 8400cc3 [0, 1, 0, -6028008, -5696952012] [2] 294912  
8400.ce5 8400cc4 [0, 1, 0, -3020008, 1973959988] [2, 2] 294912  
8400.ce2 8400cc5 [0, 1, 0, -48020008, 128063959988] [2, 2] 589824  
8400.ce8 8400cc6 [0, 1, 0, 507992, 6313399988] [2] 589824  
8400.ce1 8400cc7 [0, 1, 0, -768320008, 8196864559988] [2] 1179648  
8400.ce3 8400cc8 [0, 1, 0, -47720008, 129743359988] [2] 1179648  

Rank

sage: E.rank()
 

The elliptic curves in class 8400cc have rank \(1\).

Modular form 8400.2.a.ce

sage: E.q_eigenform(10)
 
\( q + q^{3} - q^{7} + q^{9} + 4q^{11} + 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.