# Properties

 Label 55470w Number of curves $2$ Conductor $55470$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("55470.p1")

sage: E.isogeny_class()

## Elliptic curves in class 55470w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55470.p2 55470w1 [1, 1, 1, 4584, 316533] [2] 206976 $$\Gamma_0(N)$$-optimal
55470.p1 55470w2 [1, 1, 1, -50886, 3910989] [2] 413952

## Rank

sage: E.rank()

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

## Modular form 55470.2.a.p

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.