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Properties

Label 462462.gq
Number of curves $4$
Conductor $462462$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gq1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 462462.gq have rank \(1\).

Complex multiplication

The elliptic curves in class 462462.gq do not have complex multiplication.

Modular form 462462.2.a.gq

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} - q^{12} + q^{13} + q^{16} + q^{18} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 462462.gq

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462462.gq1 462462gq4 \([1, 1, 1, -386425663, -2923653373915]\) \(30618029936661765625/3678951124992\) \(766775749701936771084288\) \([2]\) \(119439360\) \(3.6077\)  
462462.gq2 462462gq3 \([1, 1, 1, -22147903, -53581758427]\) \(-5764706497797625/2612665516032\) \(-544537965227844827086848\) \([2]\) \(59719680\) \(3.2612\)  
462462.gq3 462462gq2 \([1, 1, 1, -10675288, 7644439529]\) \(645532578015625/252306960048\) \(52586417126224393684272\) \([2]\) \(39813120\) \(3.0584\) \(\Gamma_0(N)\)-optimal*
462462.gq4 462462gq1 \([1, 1, 1, 2131352, 862042985]\) \(5137417856375/4510142208\) \(-940014573531217696512\) \([2]\) \(19906560\) \(2.7118\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 462462.gq1.