# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("kg1")

sage: E.isogeny_class()

## Elliptic curves in class 486720kg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
486720.kg2 486720kg1 $$[0, 0, 0, 387348, 257558704]$$ $$6967871/35100$$ $$-32376856513373798400$$ $$[2]$$ $$12386304$$ $$2.4259$$ $$\Gamma_0(N)$$-optimal*
486720.kg1 486720kg2 $$[0, 0, 0, -4479852, 3269382064]$$ $$10779215329/1232010$$ $$1136427663619420323840$$ $$[2]$$ $$24772608$$ $$2.7725$$ $$\Gamma_0(N)$$-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 486720kg1.

## Rank

sage: E.rank()

The elliptic curves in class 486720kg have rank $$1$$.

## Complex multiplication

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

## Modular form 486720.2.a.kg

sage: E.q_eigenform(10)

$$q + q^{5} - 2q^{7} - 4q^{11} - 8q^{17} - 6q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.