# Properties

 Label 466752.bf Number of curves $2$ Conductor $466752$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 466752.bf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
466752.bf1 466752bf1 $$[0, -1, 0, -193313, 29820609]$$ $$3047678972871625/304559880768$$ $$79838545384046592$$ $$[2]$$ $$4128768$$ $$1.9796$$ $$\Gamma_0(N)$$-optimal
466752.bf2 466752bf2 $$[0, -1, 0, 239327, 143951041]$$ $$5783051584712375/37533175779528$$ $$-9839096831548588032$$ $$[2]$$ $$8257536$$ $$2.3262$$

## Rank

sage: E.rank()

The elliptic curves in class 466752.bf have rank $$1$$.

## Complex multiplication

The elliptic curves in class 466752.bf do not have complex multiplication.

## Modular form 466752.2.a.bf

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{7} + q^{9} + q^{11} - q^{13} - q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.