# Properties

 Label 574i Number of curves $2$ Conductor $574$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 574i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
574.g2 574i1 $$[1, -1, 1, -19353, 958713]$$ $$801581275315909089/70810888830976$$ $$70810888830976$$ $$$$ $$3528$$ $$1.3978$$ $$\Gamma_0(N)$$-optimal
574.g1 574i2 $$[1, -1, 1, -9611313, -11466507927]$$ $$98191033604529537629349729/10906239337336$$ $$10906239337336$$ $$[]$$ $$24696$$ $$2.3707$$

## Rank

sage: E.rank()

The elliptic curves in class 574i have rank $$1$$.

## Complex multiplication

The elliptic curves in class 574i do not have complex multiplication.

## Modular form574.2.a.i

sage: E.q_eigenform(10)

$$q + q^{2} - 3q^{3} + q^{4} - q^{5} - 3q^{6} + q^{7} + q^{8} + 6q^{9} - q^{10} - 2q^{11} - 3q^{12} + q^{14} + 3q^{15} + q^{16} - 3q^{17} + 6q^{18} - 8q^{19} + O(q^{20})$$ ## 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 Cremona numbering.

$$\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 