Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("hz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 486720.hz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
486720.hz1 486720hz3 \([0, 0, 0, -50620908, 138625462352]\) \(31103978031362/195\) \(89935712537149440\) \([2]\) \(33030144\) \(2.8587\) \(\Gamma_0(N)\)-optimal*
486720.hz2 486720hz4 \([0, 0, 0, -4382508, 347336912]\) \(20183398562/11567205\) \(5334896531991167631360\) \([2]\) \(33030144\) \(2.8587\)  
486720.hz3 486720hz2 \([0, 0, 0, -3165708, 2163289232]\) \(15214885924/38025\) \(8768731972372070400\) \([2, 2]\) \(16515072\) \(2.5122\) \(\Gamma_0(N)\)-optimal*
486720.hz4 486720hz1 \([0, 0, 0, -123708, 59442032]\) \(-3631696/24375\) \(-1405245508392960000\) \([2]\) \(8257536\) \(2.1656\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 486720.hz1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 486720.2.a.hz

sage: E.q_eigenform(10)
 
\(q - q^{5} + 4q^{7} + 4q^{11} - 6q^{17} + O(q^{20})\)  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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.