# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("28224.fm1")

sage: E.isogeny_class()

## Elliptic curves in class 28224ft

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
28224.fm5 28224ft1 [0, 0, 0, -113484, -32494448] [2] 294912 $$\Gamma_0(N)$$-optimal
28224.fm4 28224ft2 [0, 0, 0, -2371404, -1404406640] [2, 2] 589824
28224.fm3 28224ft3 [0, 0, 0, -2935884, -685259120] [2, 2] 1179648
28224.fm1 28224ft4 [0, 0, 0, -37933644, -89925934448] [2] 1179648
28224.fm6 28224ft5 [0, 0, 0, 10893876, -5293335152] [2] 2359296
28224.fm2 28224ft6 [0, 0, 0, -25797324, 49948258192] [2] 2359296

## Rank

sage: E.rank()

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.