# Properties

 Label 223440dc Number of curves $4$ Conductor $223440$ CM no Rank $2$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("dc1")

sage: E.isogeny_class()

## Elliptic curves in class 223440dc

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
223440.ch3 223440dc1 $$[0, -1, 0, -24320, 1459200]$$ $$3301293169/22800$$ $$10987098931200$$ $$[2]$$ $$589824$$ $$1.3359$$ $$\Gamma_0(N)$$-optimal
223440.ch2 223440dc2 $$[0, -1, 0, -40000, -635648]$$ $$14688124849/8122500$$ $$3914153994240000$$ $$[2, 2]$$ $$1179648$$ $$1.6824$$
223440.ch4 223440dc3 $$[0, -1, 0, 156000, -5182848]$$ $$871257511151/527800050$$ $$-254341726545715200$$ $$[2]$$ $$2359296$$ $$2.0290$$
223440.ch1 223440dc4 $$[0, -1, 0, -486880, -130409600]$$ $$26487576322129/44531250$$ $$21459177600000000$$ $$[2]$$ $$2359296$$ $$2.0290$$

## Rank

sage: E.rank()

The elliptic curves in class 223440dc have rank $$2$$.

## Complex multiplication

The elliptic curves in class 223440dc do not have complex multiplication.

## Modular form 223440.2.a.dc

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{9} - 4q^{11} - 2q^{13} - q^{15} - 2q^{17} - q^{19} + O(q^{20})$$

## 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.