# Properties

 Label 55470.r Number of curves $4$ Conductor $55470$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("55470.r1")

sage: E.isogeny_class()

## Elliptic curves in class 55470.r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55470.r1 55470t4 [1, 1, 1, -9582481, 9847709639] [2] 5322240
55470.r2 55470t3 [1, 1, 1, -9212681, 10758748919] [2] 2661120
55470.r3 55470t2 [1, 1, 1, -2510056, -1530074881] [2] 1774080
55470.r4 55470t1 [1, 1, 1, -198806, -10196881] [2] 887040 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form 55470.2.a.r

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.