# Properties

 Label 221067.z Number of curves $4$ Conductor $221067$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 221067.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
221067.z1 221067y3 [1, -1, 0, -12985803, 18014708344] [2] 5898240
221067.z2 221067y2 [1, -1, 0, -827118, 270323455] [2, 2] 2949120
221067.z3 221067y1 [1, -1, 0, -168273, -21544880] [2] 1474560 $$\Gamma_0(N)$$-optimal
221067.z4 221067y4 [1, -1, 0, 790047, 1197605866] [2] 5898240

## Rank

sage: E.rank()

The elliptic curves in class 221067.z have rank $$0$$.

## Complex multiplication

The elliptic curves in class 221067.z do not have complex multiplication.

## Modular form 221067.2.a.z

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 2q^{5} + q^{7} - 3q^{8} - 2q^{10} - 2q^{13} + q^{14} - q^{16} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.