# Properties

 Label 221b Number of curves $2$ Conductor $221$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 221b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221.b1 221b1 $$[1, 1, 0, -59, 152]$$ $$23320116793/2873$$ $$2873$$ $$$$ $$24$$ $$-0.31302$$ $$\Gamma_0(N)$$-optimal
221.b2 221b2 $$[1, 1, 0, -54, 185]$$ $$-17923019113/8254129$$ $$-8254129$$ $$$$ $$48$$ $$0.033555$$

## Rank

sage: E.rank()

The elliptic curves in class 221b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 221b do not have complex multiplication.

## Modular form221.2.a.b

sage: E.q_eigenform(10)

$$q + q^{2} + 2q^{3} - q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 3q^{8} + q^{9} + 2q^{10} - 6q^{11} - 2q^{12} - q^{13} + 2q^{14} + 4q^{15} - q^{16} + q^{17} + q^{18} + 4q^{19} + O(q^{20})$$ ## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 