# Properties

 Label 171b Number of curves $3$ Conductor $171$ CM no Rank $1$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 171b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
171.b3 171b1 $$[0, 0, 1, 6, 0]$$ $$32768/19$$ $$-13851$$ $$[]$$ $$8$$ $$-0.51587$$ $$\Gamma_0(N)$$-optimal
171.b2 171b2 $$[0, 0, 1, -84, 315]$$ $$-89915392/6859$$ $$-5000211$$ $$$$ $$24$$ $$0.033439$$
171.b1 171b3 $$[0, 0, 1, -6924, 221760]$$ $$-50357871050752/19$$ $$-13851$$ $$$$ $$72$$ $$0.58275$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 171b do not have complex multiplication.

## 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 Cremona 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 Cremona labels. 