Properties

Label 463680ko
Number of curves $6$
Conductor $463680$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ko1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 463680ko

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.ko6 463680ko1 \([0, 0, 0, 82968, 2860216]\) \(84611246065664/53699121315\) \(-40086179265162240\) \([2]\) \(3670016\) \(1.8748\) \(\Gamma_0(N)\)-optimal*
463680.ko5 463680ko2 \([0, 0, 0, -349212, 23431984]\) \(394315384276816/208332909225\) \(2488314934477209600\) \([2, 2]\) \(7340032\) \(2.2214\) \(\Gamma_0(N)\)-optimal*
463680.ko2 463680ko3 \([0, 0, 0, -4406412, 3556441744]\) \(198048499826486404/242568272835\) \(11588879705487114240\) \([2]\) \(14680064\) \(2.5680\) \(\Gamma_0(N)\)-optimal*
463680.ko3 463680ko4 \([0, 0, 0, -3206892, -2192984624]\) \(76343005935514084/694180580625\) \(33164993709711360000\) \([2, 2]\) \(14680064\) \(2.5680\)  
463680.ko4 463680ko5 \([0, 0, 0, -938892, -5234826224]\) \(-957928673903042/123339801817575\) \(-11785301593294395801600\) \([2]\) \(29360128\) \(2.9145\)  
463680.ko1 463680ko6 \([0, 0, 0, -51197772, -141001805936]\) \(155324313723954725282/13018359375\) \(1243923609600000000\) \([2]\) \(29360128\) \(2.9145\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 463680ko1.

Rank

sage: E.rank()
 

The elliptic curves in class 463680ko have rank \(0\).

Complex multiplication

The elliptic curves in class 463680ko do not have complex multiplication.

Modular form 463680.2.a.ko

sage: E.q_eigenform(10)
 
\(q + q^{5} - q^{7} + 4q^{11} + 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20})\)  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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.