# Properties

 Label 55470.x Number of curves $4$ Conductor $55470$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("55470.x1")

sage: E.isogeny_class()

## Elliptic curves in class 55470.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55470.x1 55470bd4 [1, 1, 1, -1550425, -738640015]  1419264
55470.x2 55470bd2 [1, 1, 1, -163675, 6322085] [2, 2] 709632
55470.x3 55470bd1 [1, 1, 1, -126695, 17282957]  354816 $$\Gamma_0(N)$$-optimal
55470.x4 55470bd3 [1, 1, 1, 631395, 50527977]  1419264

## Rank

sage: E.rank()

The elliptic curves in class 55470.x have rank $$0$$.

## Modular form 55470.2.a.x

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 