# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("28224.dg1")

sage: E.isogeny_class()

## Elliptic curves in class 28224fc

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
28224.dg5 28224fc1 [0, 0, 0, -14700, -1388464] [2] 73728 $$\Gamma_0(N)$$-optimal
28224.dg4 28224fc2 [0, 0, 0, -296940, -62239408] [2] 147456
28224.dg6 28224fc3 [0, 0, 0, 126420, 29037008] [2] 221184
28224.dg3 28224fc4 [0, 0, 0, -1002540, 316696016] [2] 442368
28224.dg2 28224fc5 [0, 0, 0, -4812780, 4075624784] [2] 663552
28224.dg1 28224fc6 [0, 0, 0, -77066220, 260401928528] [2] 1327104

## Rank

sage: E.rank()

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

## Modular form 28224.2.a.dg

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.