# Properties

 Label 19350.bf Number of curves $2$ Conductor $19350$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bf1")

sage: E.isogeny_class()

## Elliptic curves in class 19350.bf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
19350.bf1 19350y2 $$[1, -1, 0, -53442, 4616716]$$ $$1481933914201/53916840$$ $$614146505625000$$ $$$$ $$110592$$ $$1.6073$$
19350.bf2 19350y1 $$[1, -1, 0, -8442, -198284]$$ $$5841725401/1857600$$ $$21159225000000$$ $$$$ $$55296$$ $$1.2608$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 19350.bf have rank $$1$$.

## Complex multiplication

The elliptic curves in class 19350.bf do not have complex multiplication.

## Modular form 19350.2.a.bf

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + 2q^{7} - q^{8} + 2q^{11} + 2q^{13} - 2q^{14} + q^{16} - 4q^{17} - 6q^{19} + O(q^{20})$$ ## 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. 