# Properties

 Label 5054b Number of curves $2$ Conductor $5054$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 5054b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5054.b2 5054b1 $$[1, -1, 1, 20148, 32163887]$$ $$53261199/26353376$$ $$-447574222639176416$$ $$[]$$ $$47880$$ $$2.0660$$ $$\Gamma_0(N)$$-optimal
5054.b1 5054b2 $$[1, -1, 1, -27347262, -55124114227]$$ $$-133179212896925841/240518168576$$ $$-4084855478516361199616$$ $$[]$$ $$335160$$ $$3.0389$$

## Rank

sage: E.rank()

The elliptic curves in class 5054b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 5054b do not have complex multiplication.

## Modular form5054.2.a.b

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} - 3 q^{9} + q^{10} - 2 q^{11} - 5 q^{13} + q^{14} + q^{16} - 3 q^{18} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 