# Properties

 Label 223440.fd Number of curves $4$ Conductor $223440$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("223440.fd1")

sage: E.isogeny_class()

## Elliptic curves in class 223440.fd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
223440.fd1 223440ct3 [0, 1, 0, -2383376, 1415446164]  2654208
223440.fd2 223440ct4 [0, 1, 0, -172496, 14613780]  2654208
223440.fd3 223440ct2 [0, 1, 0, -148976, 22074324] [2, 2] 1327104
223440.fd4 223440ct1 [0, 1, 0, -7856, 454740]  663552 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 223440.fd have rank $$0$$.

## Modular form 223440.2.a.fd

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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 