# Properties

 Label 28224.fm Number of curves $6$ Conductor $28224$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("fm1")

sage: E.isogeny_class()

## Elliptic curves in class 28224.fm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.fm1 28224ft4 $$[0, 0, 0, -37933644, -89925934448]$$ $$268498407453697/252$$ $$5665734653902848$$ $$[2]$$ $$1179648$$ $$2.7511$$
28224.fm2 28224ft6 $$[0, 0, 0, -25797324, 49948258192]$$ $$84448510979617/933897762$$ $$20996892513356009177088$$ $$[2]$$ $$2359296$$ $$3.0977$$
28224.fm3 28224ft3 $$[0, 0, 0, -2935884, -685259120]$$ $$124475734657/63011844$$ $$1416699953004445433856$$ $$[2, 2]$$ $$1179648$$ $$2.7511$$
28224.fm4 28224ft2 $$[0, 0, 0, -2371404, -1404406640]$$ $$65597103937/63504$$ $$1427765132783517696$$ $$[2, 2]$$ $$589824$$ $$2.4045$$
28224.fm5 28224ft1 $$[0, 0, 0, -113484, -32494448]$$ $$-7189057/16128$$ $$-362607017849782272$$ $$[2]$$ $$294912$$ $$2.0579$$ $$\Gamma_0(N)$$-optimal
28224.fm6 28224ft5 $$[0, 0, 0, 10893876, -5293335152]$$ $$6359387729183/4218578658$$ $$-94846616241450678484992$$ $$[2]$$ $$2359296$$ $$3.0977$$

## Rank

sage: E.rank()

The elliptic curves in class 28224.fm have rank $$0$$.

## Complex multiplication

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

## Modular form 28224.2.a.fm

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.