Properties

Label 486720je
Number of curves $4$
Conductor $486720$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("je1")
 
E.isogeny_class()
 

Elliptic curves in class 486720je

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
486720.je4 486720je1 \([0, 0, 0, -28085772, -94645951504]\) \(-2656166199049/2658140160\) \(-2451915172443204291133440\) \([2]\) \(82575360\) \(3.3756\) \(\Gamma_0(N)\)-optimal*
486720.je3 486720je2 \([0, 0, 0, -526487052, -4648239406096]\) \(17496824387403529/6580454400\) \(6069926720843284060569600\) \([2, 2]\) \(165150720\) \(3.7222\) \(\Gamma_0(N)\)-optimal*
486720.je2 486720je3 \([0, 0, 0, -604362252, -3182721592336]\) \(26465989780414729/10571870144160\) \(9751678710412783278033469440\) \([2]\) \(330301440\) \(4.0688\) \(\Gamma_0(N)\)-optimal*
486720.je1 486720je4 \([0, 0, 0, -8423032332, -297543738313744]\) \(71647584155243142409/10140000\) \(9353314103863541760000\) \([2]\) \(330301440\) \(4.0688\)  
*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 486720je1.

Rank

sage: E.rank()
 

The elliptic curves in class 486720je have rank \(1\).

Complex multiplication

The elliptic curves in class 486720je do not have complex multiplication.

Modular form 486720.2.a.je

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.