# Properties

 Label 28224.ew Number of curves $2$ Conductor $28224$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("ew1")

sage: E.isogeny_class()

## Elliptic curves in class 28224.ew

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.ew1 28224bw2 $$[0, 0, 0, -448644, 115632160]$$ $$82881856/27$$ $$3253371070795776$$ $$[2]$$ $$172032$$ $$1.9504$$
28224.ew2 28224bw1 $$[0, 0, 0, -31899, 1277332]$$ $$1906624/729$$ $$1372515920491968$$ $$[2]$$ $$86016$$ $$1.6038$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 28224.ew have rank $$1$$.

## Complex multiplication

The elliptic curves in class 28224.ew do not have complex multiplication.

## Modular form 28224.2.a.ew

sage: E.q_eigenform(10)

$$q + 2q^{5} - 2q^{11} - 2q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.