# Properties

 Label 55470s Number of curves $2$ Conductor $55470$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("55470.u1")

sage: E.isogeny_class()

## Elliptic curves in class 55470s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55470.u2 55470s1 [1, 1, 1, -69376, -4749727]  532224 $$\Gamma_0(N)$$-optimal
55470.u1 55470s2 [1, 1, 1, -439176, 108261153]  1064448

## Rank

sage: E.rank()

The elliptic curves in class 55470s have rank $$1$$.

## Modular form 55470.2.a.u

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + 2q^{7} + q^{8} + q^{9} - q^{10} - 2q^{11} - q^{12} - 2q^{13} + 2q^{14} + q^{15} + q^{16} - 4q^{17} + q^{18} + 6q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 