# Properties

 Label 28224ce Number of curves $4$ Conductor $28224$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("ce1")

sage: E.isogeny_class()

## Elliptic curves in class 28224ce

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
28224.br4 28224ce1 $$[0, 0, 0, -12936, 3674216]$$ $$-2725888/64827$$ $$-5693399373892608$$ $$[2]$$ $$147456$$ $$1.7039$$ $$\Gamma_0(N)$$-optimal
28224.br3 28224ce2 $$[0, 0, 0, -445116, 113793680]$$ $$6940769488/35721$$ $$50194867949420544$$ $$[2, 2]$$ $$294912$$ $$2.0504$$
28224.br2 28224ce3 $$[0, 0, 0, -692076, -26578384]$$ $$6522128932/3720087$$ $$20909747848644329472$$ $$[2]$$ $$589824$$ $$2.3970$$
28224.br1 28224ce4 $$[0, 0, 0, -7113036, 7301811440]$$ $$7080974546692/189$$ $$1062325247606784$$ $$[2]$$ $$589824$$ $$2.3970$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 28224.2.a.ce

sage: E.q_eigenform(10)

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

## 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.