# Properties

 Label 28224gd Number of curves $4$ Conductor $28224$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("gd1")

sage: E.isogeny_class()

## Elliptic curves in class 28224gd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
28224.by4 28224gd1 [0, 0, 0, 1764, -148176] [2] 49152 $$\Gamma_0(N)$$-optimal
28224.by3 28224gd2 [0, 0, 0, -33516, -2222640] [2, 2] 98304
28224.by2 28224gd3 [0, 0, 0, -104076, 10224144] [2] 196608
28224.by1 28224gd4 [0, 0, 0, -527436, -147435120] [2] 196608

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 28224.2.a.gd

sage: E.q_eigenform(10)

$$q - 2q^{5} + 4q^{11} + 2q^{13} - 6q^{17} - 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.