# Properties

 Label 28224bz Number of curves 6 Conductor 28224 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("28224.fk1")
sage: E.isogeny_class()

## Elliptic curves in class 28224bz

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
28224.fk6 28224bz1 [0, 0, 0, 27636, -389648] 2 98304 $$\Gamma_0(N)$$-optimal
28224.fk5 28224bz2 [0, 0, 0, -113484, -3155600] 4 196608
28224.fk3 28224bz3 [0, 0, 0, -1101324, 442162672] 2 393216
28224.fk2 28224bz4 [0, 0, 0, -1383564, -625494800] 4 393216
28224.fk4 28224bz5 [0, 0, 0, -960204, -1015494032] 2 786432
28224.fk1 28224bz6 [0, 0, 0, -22128204, -40065204368] 2 786432

## Rank

sage: E.rank()

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

## Modular form 28224.2.a.fk

sage: E.q_eigenform(10)
$$q + 2q^{5} + 4q^{11} - 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.