# Properties

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

# Related objects

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)

$$q - q^{5} + 4q^{7} + 4q^{11} - 6q^{17} + O(q^{20})$$

## 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.