# Properties

 Label 221067.g Number of curves $2$ Conductor $221067$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

## Elliptic curves in class 221067.g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221067.g1 221067f1 $$[1, -1, 1, -320, 938]$$ $$3723875/1827$$ $$1772736273$$ $$$$ $$79872$$ $$0.46791$$ $$\Gamma_0(N)$$-optimal
221067.g2 221067f2 $$[1, -1, 1, 1165, 6284]$$ $$180362125/123627$$ $$-119955154473$$ $$$$ $$159744$$ $$0.81448$$

## Rank

sage: E.rank()

The elliptic curves in class 221067.g have rank $$1$$.

## Complex multiplication

The elliptic curves in class 221067.g do not have complex multiplication.

## Modular form 221067.2.a.g

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 