# Properties

 Label 55.a Number of curves $4$ Conductor $55$ CM no Rank $0$ Graph # Learn more

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 55.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
55.a1 55a3 $$[1, -1, 0, -59, 190]$$ $$22930509321/6875$$ $$6875$$ $$$$ $$4$$ $$-0.28567$$
55.a2 55a2 $$[1, -1, 0, -29, -52]$$ $$2749884201/73205$$ $$73205$$ $$$$ $$4$$ $$-0.28567$$
55.a3 55a1 $$[1, -1, 0, -4, 3]$$ $$8120601/3025$$ $$3025$$ $$[2, 2]$$ $$2$$ $$-0.63224$$ $$\Gamma_0(N)$$-optimal
55.a4 55a4 $$[1, -1, 0, 1, 0]$$ $$59319/55$$ $$-55$$ $$$$ $$4$$ $$-0.97881$$

## Rank

sage: E.rank()

The elliptic curves in class 55.a have rank $$0$$.

## Complex multiplication

The elliptic curves in class 55.a do not have complex multiplication.

## Modular form55.2.a.a

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} + q^{5} - 3q^{8} - 3q^{9} + q^{10} - q^{11} + 2q^{13} - q^{16} + 6q^{17} - 3q^{18} - 4q^{19} + O(q^{20})$$ ## 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. 