# Properties

 Label 5610.bf Number of curves $4$ Conductor $5610$ CM no Rank $1$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 5610.bf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5610.bf1 5610bj4 $$[1, 0, 0, -84486, 9444996]$$ $$66692696957462376289/1322972640$$ $$1322972640$$ $$[2]$$ $$20480$$ $$1.2827$$
5610.bf2 5610bj3 $$[1, 0, 0, -8006, -21180]$$ $$56751044592329569/32660264340000$$ $$32660264340000$$ $$[2]$$ $$20480$$ $$1.2827$$
5610.bf3 5610bj2 $$[1, 0, 0, -5286, 146916]$$ $$16334668434139489/72511718400$$ $$72511718400$$ $$[2, 2]$$ $$10240$$ $$0.93612$$
5610.bf4 5610bj1 $$[1, 0, 0, -166, 4580]$$ $$-506071034209/8823767040$$ $$-8823767040$$ $$[2]$$ $$5120$$ $$0.58954$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 5610.bf have rank $$1$$.

## Complex multiplication

The elliptic curves in class 5610.bf do not have complex multiplication.

## Modular form5610.2.a.bf

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - 4 q^{7} + q^{8} + q^{9} - q^{10} + q^{11} + q^{12} - 2 q^{13} - 4 q^{14} - q^{15} + q^{16} + q^{17} + q^{18} + O(q^{20})$$

## 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.