# Properties

 Label 450840.w Number of curves $4$ Conductor $450840$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("w1")

sage: E.isogeny_class()

## Elliptic curves in class 450840.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.w1 450840w4 $$[0, -1, 0, -9449240, -11176780980]$$ $$1887517194957938/21849165$$ $$1080085970493204480$$ $$[2]$$ $$14155776$$ $$2.6115$$
450840.w2 450840w2 $$[0, -1, 0, -605840, -164979300]$$ $$994958062276/98903025$$ $$2444573276412134400$$ $$[2, 2]$$ $$7077888$$ $$2.2649$$
450840.w3 450840w1 $$[0, -1, 0, -137660, 16861812]$$ $$46689225424/7249905$$ $$44798741038321920$$ $$[2]$$ $$3538944$$ $$1.9183$$ $$\Gamma_0(N)$$-optimal*
450840.w4 450840w3 $$[0, -1, 0, 746680, -798499668]$$ $$931329171502/6107473125$$ $$-301915246531242240000$$ $$[2]$$ $$14155776$$ $$2.6115$$
*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 450840.w1.

## Rank

sage: E.rank()

The elliptic curves in class 450840.w have rank $$0$$.

## Complex multiplication

The elliptic curves in class 450840.w do not have complex multiplication.

## Modular form 450840.2.a.w

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{9} - 4q^{11} - q^{13} - q^{15} + 8q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.