Properties

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

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Show commands: SageMath
E = EllipticCurve("cn1")
 
E.isogeny_class()
 

Elliptic curves in class 8400.cn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8400.cn1 8400ce7 \([0, 1, 0, -2580408, -1004608812]\) \(29689921233686449/10380965400750\) \(664381785648000000000\) \([2]\) \(331776\) \(2.6969\)  
8400.cn2 8400ce4 \([0, 1, 0, -2304408, -1347208812]\) \(21145699168383889/2593080\) \(165957120000000\) \([2]\) \(110592\) \(2.1476\)  
8400.cn3 8400ce6 \([0, 1, 0, -1080408, 420391188]\) \(2179252305146449/66177562500\) \(4235364000000000000\) \([2, 2]\) \(165888\) \(2.3503\)  
8400.cn4 8400ce3 \([0, 1, 0, -1072408, 427095188]\) \(2131200347946769/2058000\) \(131712000000000\) \([2]\) \(82944\) \(2.0037\)  
8400.cn5 8400ce2 \([0, 1, 0, -144408, -20968812]\) \(5203798902289/57153600\) \(3657830400000000\) \([2, 2]\) \(55296\) \(1.8010\)  
8400.cn6 8400ce5 \([0, 1, 0, -32408, -52552812]\) \(-58818484369/18600435000\) \(-1190427840000000000\) \([2]\) \(110592\) \(2.1476\)  
8400.cn7 8400ce1 \([0, 1, 0, -16408, 279188]\) \(7633736209/3870720\) \(247726080000000\) \([2]\) \(27648\) \(1.4544\) \(\Gamma_0(N)\)-optimal
8400.cn8 8400ce8 \([0, 1, 0, 291592, 1416463188]\) \(42841933504271/13565917968750\) \(-868218750000000000000\) \([2]\) \(331776\) \(2.6969\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 8400.cn do not have complex multiplication.

Modular form 8400.2.a.cn

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{7} + q^{9} - 2 q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.