# Properties

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

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 54150.by

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54150.by1 54150bu4 $$[1, 1, 1, -5604713, 5097385781]$$ $$26487576322129/44531250$$ $$32734560754394531250$$ $$[2]$$ $$2211840$$ $$2.6399$$
54150.by2 54150bu2 $$[1, 1, 1, -460463, 25155281]$$ $$14688124849/8122500$$ $$5970783881601562500$$ $$[2, 2]$$ $$1105920$$ $$2.2933$$
54150.by3 54150bu1 $$[1, 1, 1, -279963, -56791719]$$ $$3301293169/22800$$ $$16760095106250000$$ $$[2]$$ $$552960$$ $$1.9467$$ $$\Gamma_0(N)$$-optimal
54150.by4 54150bu3 $$[1, 1, 1, 1795787, 201142781]$$ $$871257511151/527800050$$ $$-387981536626469531250$$ $$[2]$$ $$2211840$$ $$2.6399$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 54150.by do not have complex multiplication.

## 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.