# Properties

 Label 485100.he Number of curves $4$ Conductor $485100$ CM no Rank $1$ Graph

# Learn more

Show commands for: SageMath
sage: E = EllipticCurve("he1")

sage: E.isogeny_class()

## Elliptic curves in class 485100.he

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485100.he1 485100he4 $$[0, 0, 0, -546799575, 2594830414750]$$ $$52702650535889104/22020583921875$$ $$7554480260536743187500000000$$ $$[2]$$ $$286654464$$ $$4.0464$$
485100.he2 485100he2 $$[0, 0, 0, -471388575, 3939277567750]$$ $$33766427105425744/9823275$$ $$3370016769065100000000$$ $$[2]$$ $$95551488$$ $$3.4971$$ $$\Gamma_0(N)$$-optimal*
485100.he3 485100he1 $$[0, 0, 0, -29341200, 62080041625]$$ $$-130287139815424/2250652635$$ $$-48257436555594708750000$$ $$[2]$$ $$47775744$$ $$3.1505$$ $$\Gamma_0(N)$$-optimal*
485100.he4 485100he3 $$[0, 0, 0, 113542800, 297499292125]$$ $$7549996227362816/6152409907875$$ $$-131917083150101525718750000$$ $$[2]$$ $$143327232$$ $$3.6998$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 485100.he1.

## Rank

sage: E.rank()

The elliptic curves in class 485100.he have rank $$1$$.

## Complex multiplication

The elliptic curves in class 485100.he do not have complex multiplication.

## Modular form 485100.2.a.he

sage: E.q_eigenform(10)

$$q + q^{11} + 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.