# Properties

 Label 57222x Number of curves $2$ Conductor $57222$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 57222x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
57222.bd1 57222x1 [1, -1, 0, -509850, 140162548] [2] 884736 $$\Gamma_0(N)$$-optimal
57222.bd2 57222x2 [1, -1, 0, -405810, 198945148] [2] 1769472

## Rank

sage: E.rank()

The elliptic curves in class 57222x have rank $$1$$.

## Complex multiplication

The elliptic curves in class 57222x do not have complex multiplication.

## Modular form 57222.2.a.x

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + 4q^{5} + 2q^{7} - q^{8} - 4q^{10} + q^{11} - 2q^{14} + q^{16} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.