# Properties

 Label 55470.d Number of curves $4$ Conductor $55470$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("55470.d1")

sage: E.isogeny_class()

## Elliptic curves in class 55470.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55470.d1 55470c3 [1, 1, 0, -63431832, 84443489676]  15966720
55470.d2 55470c1 [1, 1, 0, -32479572, -71256562416]  5322240 $$\Gamma_0(N)$$-optimal
55470.d3 55470c2 [1, 1, 0, -30630572, -79724612616]  10644480
55470.d4 55470c4 [1, 1, 0, 225474418, 639663520926]  31933440

## Rank

sage: E.rank()

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

## Modular form 55470.2.a.d

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 