# Properties

 Label 221067.k Number of curves $6$ Conductor $221067$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("221067.k1")

sage: E.isogeny_class()

## Elliptic curves in class 221067.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
221067.k1 221067l3 [1, -1, 1, -222835559, 1280394355056] [2] 15728640
221067.k2 221067l6 [1, -1, 1, -46248764, -98399914104] [2] 31457280
221067.k3 221067l4 [1, -1, 1, -14194049, 19202424288] [2, 2] 15728640
221067.k4 221067l2 [1, -1, 1, -13927244, 20008708998] [2, 2] 7864320
221067.k5 221067l1 [1, -1, 1, -853799, 325330206] [2] 3932160 $$\Gamma_0(N)$$-optimal
221067.k6 221067l5 [1, -1, 1, 13591786, 85199339580] [2] 31457280

## Rank

sage: E.rank()

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

## Modular form 221067.2.a.k

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.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.