Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("mq1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 486720mq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
486720.mq3 486720mq1 \([0, 0, 0, -395967, -95903444]\) \(30488290624/195\) \(43913922137280\) \([2]\) \(2064384\) \(1.8024\) \(\Gamma_0(N)\)-optimal*
486720.mq2 486720mq2 \([0, 0, 0, -403572, -92027936]\) \(504358336/38025\) \(548045748273254400\) \([2, 2]\) \(4128768\) \(2.1490\) \(\Gamma_0(N)\)-optimal*
486720.mq1 486720mq3 \([0, 0, 0, -1316172, 472688944]\) \(2186875592/428415\) \(49397190111029329920\) \([2]\) \(8257536\) \(2.4956\) \(\Gamma_0(N)\)-optimal*
486720.mq4 486720mq4 \([0, 0, 0, 387348, -408712304]\) \(55742968/658125\) \(-75883257453219840000\) \([2]\) \(8257536\) \(2.4956\)  
*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 486720mq1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 486720.2.a.mq

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.