# Properties

 Label 28224.dn Number of curves $2$ Conductor $28224$ CM $$\Q(\sqrt{-3})$$ Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("dn1")

sage: E.isogeny_class()

## Elliptic curves in class 28224.dn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
28224.dn1 28224dj2 $$[0, 0, 0, 0, -6048]$$ $$0$$ $$-15801827328$$ $$[]$$ $$13824$$ $$0.63583$$   $$-3$$
28224.dn2 28224dj1 $$[0, 0, 0, 0, 224]$$ $$0$$ $$-21676032$$ $$[]$$ $$4608$$ $$0.086526$$ $$\Gamma_0(N)$$-optimal $$-3$$

## Rank

sage: E.rank()

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

## Complex multiplication

Each elliptic curve in class 28224.dn has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form 28224.2.a.dn

sage: E.q_eigenform(10)

$$q + 5q^{13} + q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 