# Properties

 Label 121b Number of curves 2 Conductor 121 CM -11 Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("121.b1")

sage: E.isogeny_class()

## Elliptic curves in class 121b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
121.b2 121b1 [0, -1, 1, -7, 10] [] 4 $$\Gamma_0(N)$$-optimal
121.b1 121b2 [0, -1, 1, -887, -10143] [] 44

## Rank

sage: E.rank()

The elliptic curves in class 121b have rank $$1$$.

## Modular form121.2.a.b

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{4} - 3q^{5} - 2q^{9} + 2q^{12} + 3q^{15} + 4q^{16} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 