# Properties

 Label 54208.bj Number of curves $4$ Conductor $54208$ CM no Rank $1$ Graph # Learn more

Show commands: SageMath
sage: E = EllipticCurve("bj1")

sage: E.isogeny_class()

## Elliptic curves in class 54208.bj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54208.bj1 54208bx4 $$[0, 0, 0, -144716, 21189520]$$ $$1443468546/7$$ $$1625414303744$$ $$$$ $$163840$$ $$1.5435$$
54208.bj2 54208bx3 $$[0, 0, 0, -28556, -1469424]$$ $$11090466/2401$$ $$557517106184192$$ $$$$ $$163840$$ $$1.5435$$
54208.bj3 54208bx2 $$[0, 0, 0, -9196, 319440]$$ $$740772/49$$ $$5688950063104$$ $$[2, 2]$$ $$81920$$ $$1.1969$$
54208.bj4 54208bx1 $$[0, 0, 0, 484, 21296]$$ $$432/7$$ $$-203176787968$$ $$$$ $$40960$$ $$0.85034$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 54208.bj have rank $$1$$.

## Complex multiplication

The elliptic curves in class 54208.bj do not have complex multiplication.

## Modular form 54208.2.a.bj

sage: E.q_eigenform(10)

$$q - 2q^{5} - q^{7} - 3q^{9} + 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 