# Properties

 Label 171.b Number of curves 3 Conductor 171 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("171.b1")

sage: E.isogeny_class()

## Elliptic curves in class 171.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
171.b1 171b3 [0, 0, 1, -6924, 221760]  72
171.b2 171b2 [0, 0, 1, -84, 315]  24
171.b3 171b1 [0, 0, 1, 6, 0] [] 8 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 171.b have rank $$1$$.

## Modular form171.2.a.b

sage: E.q_eigenform(10)

$$q - 2q^{4} - 3q^{5} - q^{7} - 3q^{11} - 4q^{13} + 4q^{16} + 3q^{17} + q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 