# Properties

 Label 223440.ch Number of curves $4$ Conductor $223440$ CM no Rank $2$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("223440.ch1")

sage: E.isogeny_class()

## Elliptic curves in class 223440.ch

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
223440.ch1 223440dc4 [0, -1, 0, -486880, -130409600]  2359296
223440.ch2 223440dc2 [0, -1, 0, -40000, -635648] [2, 2] 1179648
223440.ch3 223440dc1 [0, -1, 0, -24320, 1459200]  589824 $$\Gamma_0(N)$$-optimal
223440.ch4 223440dc3 [0, -1, 0, 156000, -5182848]  2359296

## Rank

sage: E.rank()

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

## Modular form 223440.2.a.ch

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 LMFDB 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 LMFDB labels. 