# Properties

 Label 446160.dw Number of curves $4$ Conductor $446160$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("dw1")

sage: E.isogeny_class()

## Elliptic curves in class 446160.dw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
446160.dw1 446160dw4 $$[0, -1, 0, -42249380, 105714969372]$$ $$6749703004355978704/5671875$$ $$7008526668000000$$ $$[2]$$ $$15925248$$ $$2.7760$$ $$\Gamma_0(N)$$-optimal*
446160.dw2 446160dw3 $$[0, -1, 0, -2640005, 1653219372]$$ $$-26348629355659264/24169921875$$ $$-1866617542968750000$$ $$[2]$$ $$7962624$$ $$2.4295$$ $$\Gamma_0(N)$$-optimal*
446160.dw3 446160dw2 $$[0, -1, 0, -533420, 138254700]$$ $$13584145739344/1195803675$$ $$1477610480825107200$$ $$[2]$$ $$5308416$$ $$2.2267$$ $$\Gamma_0(N)$$-optimal*
446160.dw4 446160dw1 $$[0, -1, 0, 36955, 10034400]$$ $$72268906496/606436875$$ $$-46834479458910000$$ $$[2]$$ $$2654208$$ $$1.8801$$ $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 446160.dw1.

## Rank

sage: E.rank()

The elliptic curves in class 446160.dw have rank $$0$$.

## Complex multiplication

The elliptic curves in class 446160.dw do not have complex multiplication.

## Modular form 446160.2.a.dw

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + 2q^{7} + q^{9} + q^{11} - q^{15} + 2q^{19} + O(q^{20})$$

## 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 LMFDB 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 LMFDB labels.