# Properties

 Label 171.a Number of curves $4$ Conductor $171$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 171.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
171.a1 171a3 $$[1, -1, 1, -914, -10402]$$ $$115714886617/1539$$ $$1121931$$ $$$$ $$48$$ $$0.30430$$
171.a2 171a2 $$[1, -1, 1, -59, -142]$$ $$30664297/3249$$ $$2368521$$ $$[2, 2]$$ $$24$$ $$-0.042276$$
171.a3 171a1 $$[1, -1, 1, -14, 20]$$ $$389017/57$$ $$41553$$ $$$$ $$12$$ $$-0.38885$$ $$\Gamma_0(N)$$-optimal
171.a4 171a4 $$[1, -1, 1, 76, -790]$$ $$67419143/390963$$ $$-285012027$$ $$$$ $$48$$ $$0.30430$$

## Rank

sage: E.rank()

The elliptic curves in class 171.a have rank $$0$$.

## Complex multiplication

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

## Modular form171.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + 2 q^{5} + 3 q^{8} - 2 q^{10} + 6 q^{13} - q^{16} + 6 q^{17} - q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 