# Properties

 Label 120666p Number of curves $4$ Conductor $120666$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 120666p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
120666.p4 120666p1 $$[1, 1, 0, 166, 221460]$$ $$103823/4386816$$ $$-21174322950144$$ $$$$ $$368640$$ $$1.2359$$ $$\Gamma_0(N)$$-optimal
120666.p3 120666p2 $$[1, 1, 0, -53914, 4710100]$$ $$3590714269297/73410624$$ $$354339060618816$$ $$[2, 2]$$ $$737280$$ $$1.5825$$
120666.p2 120666p3 $$[1, 1, 0, -114754, -7956788]$$ $$34623662831857/14438442312$$ $$69691603297542408$$ $$$$ $$1474560$$ $$1.9290$$
120666.p1 120666p4 $$[1, 1, 0, -858354, 305731548]$$ $$14489843500598257/6246072$$ $$30148596544248$$ $$$$ $$1474560$$ $$1.9290$$

## Rank

sage: E.rank()

The elliptic curves in class 120666p have rank $$1$$.

## Complex multiplication

The elliptic curves in class 120666p do not have complex multiplication.

## Modular form 120666.2.a.p

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - 2q^{10} - q^{12} - q^{14} - 2q^{15} + q^{16} + q^{17} - q^{18} + O(q^{20})$$ ## 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 Cremona 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 Cremona labels. 