Properties

Label 8400.cm
Number of curves 8
Conductor 8400
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("8400.cm1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8400.cm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8400.cm1 8400cf7 [0, 1, 0, -140493008, 640912067988] [2] 663552  
8400.cm2 8400cf6 [0, 1, 0, -8781008, 10011587988] [2, 2] 331776  
8400.cm3 8400cf8 [0, 1, 0, -8141008, 11533507988] [4] 663552  
8400.cm4 8400cf4 [0, 1, 0, -1743008, 869567988] [2] 221184  
8400.cm5 8400cf3 [0, 1, 0, -589008, 132035988] [2] 165888  
8400.cm6 8400cf2 [0, 1, 0, -231008, -22512012] [2, 2] 110592  
8400.cm7 8400cf1 [0, 1, 0, -199008, -34224012] [2] 55296 \(\Gamma_0(N)\)-optimal
8400.cm8 8400cf5 [0, 1, 0, 768992, -164512012] [4] 221184  

Rank

sage: E.rank()
 

The elliptic curves in class 8400.cm have rank \(0\).

Modular form 8400.2.a.cm

sage: E.q_eigenform(10)
 
\( q + q^{3} + q^{7} + q^{9} - 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.