# Properties

 Label 161.a Number of curves $4$ Conductor $161$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 161.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
161.a1 161a3 $$[1, -1, 1, -124, 560]$$ $$209267191953/55223$$ $$55223$$ $$$$ $$20$$ $$-0.10563$$
161.a2 161a1 $$[1, -1, 1, -9, 8]$$ $$72511713/25921$$ $$25921$$ $$[2, 2]$$ $$10$$ $$-0.45221$$ $$\Gamma_0(N)$$-optimal
161.a3 161a2 $$[1, -1, 1, -4, -2]$$ $$5545233/161$$ $$161$$ $$$$ $$20$$ $$-0.79878$$
161.a4 161a4 $$[1, -1, 1, 26, 36]$$ $$2014698447/1958887$$ $$-1958887$$ $$$$ $$20$$ $$-0.10563$$

## Rank

sage: E.rank()

The elliptic curves in class 161.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 161.a do not have complex multiplication.

## Modular form161.2.a.a

sage: E.q_eigenform(10)

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