# Properties

 Label 28224.fn Number of curves $2$ Conductor $28224$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("fn1")

sage: E.isogeny_class()

## Elliptic curves in class 28224.fn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.fn1 28224du1 $$[0, 0, 0, -10584, -370440]$$ $$55296/7$$ $$16598831993856$$ $$$$ $$73728$$ $$1.2659$$ $$\Gamma_0(N)$$-optimal
28224.fn2 28224du2 $$[0, 0, 0, 15876, -1926288]$$ $$11664/49$$ $$-1859069183311872$$ $$$$ $$147456$$ $$1.6125$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 28224.2.a.fn

sage: E.q_eigenform(10)

$$q + 2q^{5} + 6q^{11} - 6q^{13} + 2q^{17} - 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 