# Properties

 Label 2166.a Number of curves $4$ Conductor $2166$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 2166.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2166.a1 2166b3 $$[1, 1, 0, -154515, 23313369]$$ $$8671983378625/82308$$ $$3872252373348$$ $$$$ $$12960$$ $$1.5777$$
2166.a2 2166b4 $$[1, 1, 0, -150905, 24459183]$$ $$-8078253774625/846825858$$ $$-39839668543190898$$ $$$$ $$25920$$ $$1.9243$$
2166.a3 2166b1 $$[1, 1, 0, -2895, -5787]$$ $$57066625/32832$$ $$1544610364992$$ $$$$ $$4320$$ $$1.0284$$ $$\Gamma_0(N)$$-optimal
2166.a4 2166b2 $$[1, 1, 0, 11545, -31779]$$ $$3616805375/2105352$$ $$-99048139655112$$ $$$$ $$8640$$ $$1.3750$$

## Rank

sage: E.rank()

The elliptic curves in class 2166.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 2166.a do not have complex multiplication.

## Modular form2166.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{6} - 4q^{7} - q^{8} + q^{9} - q^{12} + 4q^{13} + 4q^{14} + q^{16} + 6q^{17} - q^{18} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 