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Properties

Label 486720.dq
Number of curves $2$
Conductor $486720$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dq1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

Complex multiplication

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

Modular form 486720.2.a.dq

Copy content sage:E.q_eigenform(10)
 
\(q - q^{5} - 6 q^{11} + 4 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 486720.dq

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
486720.dq1 486720dq2 \([0, 0, 0, -123708, -15502032]\) \(98055792/8125\) \(17348709980160000\) \([2]\) \(4128768\) \(1.8582\) \(\Gamma_0(N)\)-optimal*
486720.dq2 486720dq1 \([0, 0, 0, 8112, -1107288]\) \(442368/4225\) \(-563833074355200\) \([2]\) \(2064384\) \(1.5117\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 486720.dq1.