# Properties

 Label 485100cn Number of curves $4$ Conductor $485100$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("cn1")

sage: E.isogeny_class()

## Elliptic curves in class 485100cn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485100.cn4 485100cn1 $$[0, 0, 0, 2410800, -5293733375]$$ $$72268906496/606436875$$ $$-13002934600027968750000$$ $$[2]$$ $$19906560$$ $$2.9247$$ $$\Gamma_0(N)$$-optimal*
485100.cn3 485100cn2 $$[0, 0, 0, -34798575, -72754330250]$$ $$13584145739344/1195803675$$ $$410237770729178700000000$$ $$[2]$$ $$39813120$$ $$3.2712$$ $$\Gamma_0(N)$$-optimal*
485100.cn2 485100cn3 $$[0, 0, 0, -172225200, -870636942875]$$ $$-26348629355659264/24169921875$$ $$-518240111022949218750000$$ $$[2]$$ $$59719680$$ $$3.4740$$
485100.cn1 485100cn4 $$[0, 0, 0, -2756209575, -55695033427250]$$ $$6749703004355978704/5671875$$ $$1945818870187500000000$$ $$[2]$$ $$119439360$$ $$3.8205$$
*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 485100cn1.

## Rank

sage: E.rank()

The elliptic curves in class 485100cn have rank $$0$$.

## Complex multiplication

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

## Modular form 485100.2.a.cn

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.