# Properties

 Label 54.b Number of curves $3$ Conductor $54$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 54.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
54.b1 54b2 [1, -1, 1, -29, -53] [] 6
54.b2 54b3 [1, -1, 1, -14, 29] [9] 6
54.b3 54b1 [1, -1, 1, 1, -1] [3] 2 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 54.b have rank $$0$$.

## Complex multiplication

The elliptic curves in class 54.b do not have complex multiplication.

## Modular form54.2.a.b

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - 3q^{5} - q^{7} + q^{8} - 3q^{10} + 3q^{11} - 4q^{13} - q^{14} + q^{16} + 2q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.