# Properties

 Label 54150.by Number of curves $4$ Conductor $54150$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("54150.by1")

sage: E.isogeny_class()

## Elliptic curves in class 54150.by

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
54150.by1 54150bu4 [1, 1, 1, -5604713, 5097385781]  2211840
54150.by2 54150bu2 [1, 1, 1, -460463, 25155281] [2, 2] 1105920
54150.by3 54150bu1 [1, 1, 1, -279963, -56791719]  552960 $$\Gamma_0(N)$$-optimal
54150.by4 54150bu3 [1, 1, 1, 1795787, 201142781]  2211840

## Rank

sage: E.rank()

The elliptic curves in class 54150.by have rank $$1$$.

## Modular form 54150.2.a.by

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} + 4q^{11} - q^{12} + 2q^{13} + q^{16} - 2q^{17} + q^{18} + O(q^{20})$$

## 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 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. 