# Properties

 Label 19110.df Number of curves $2$ Conductor $19110$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("df1")

sage: E.isogeny_class()

## Elliptic curves in class 19110.df

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
19110.df1 19110di2 $$[1, 0, 0, -2255, 36735]$$ $$10779215329/1232010$$ $$144944744490$$ $$[2]$$ $$34560$$ $$0.87391$$
19110.df2 19110di1 $$[1, 0, 0, 195, 2925]$$ $$6967871/35100$$ $$-4129479900$$ $$[2]$$ $$17280$$ $$0.52734$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 19110.df have rank $$0$$.

## Complex multiplication

The elliptic curves in class 19110.df do not have complex multiplication.

## Modular form 19110.2.a.df

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} + 4q^{11} + q^{12} + q^{13} + q^{15} + q^{16} - 8q^{17} + q^{18} + 6q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.