# Properties

 Label 550.d Number of curves $2$ Conductor $550$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("d1")

E.isogeny_class()

## Elliptic curves in class 550.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
550.d1 550a1 $$[1, 1, 0, -25, 125]$$ $$-117649/440$$ $$-6875000$$ $$[]$$ $$96$$ $$-0.0023865$$ $$\Gamma_0(N)$$-optimal
550.d2 550a2 $$[1, 1, 0, 225, -3125]$$ $$80062991/332750$$ $$-5199218750$$ $$[]$$ $$288$$ $$0.54692$$

## Rank

sage: E.rank()

The elliptic curves in class 550.d have rank $$1$$.

## Complex multiplication

The elliptic curves in class 550.d do not have complex multiplication.

## Modular form550.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{6} + q^{7} - q^{8} - 2 q^{9} - q^{11} - q^{12} - 2 q^{13} - q^{14} + q^{16} + 3 q^{17} + 2 q^{18} - q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.