# Properties

 Label 5520.bf Number of curves $6$ Conductor $5520$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 5520.bf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5520.bf1 5520k5 $$[0, 1, 0, -870880, 301514900]$$ $$35667215800077781442/1427217706746225$$ $$2922941863416268800$$ $$[4]$$ $$86016$$ $$2.3097$$
5520.bf2 5520k3 $$[0, 1, 0, -141880, -14287900]$$ $$308453964046598884/92949363050625$$ $$95180147763840000$$ $$[2, 4]$$ $$43008$$ $$1.9631$$
5520.bf3 5520k2 $$[0, 1, 0, -129380, -17952900]$$ $$935596404100595536/150641015625$$ $$38564100000000$$ $$[2, 2]$$ $$21504$$ $$1.6165$$
5520.bf4 5520k1 $$[0, 1, 0, -129375, -17954352]$$ $$14967807005098080256/388125$$ $$6210000$$ $$[2]$$ $$10752$$ $$1.2699$$ $$\Gamma_0(N)$$-optimal
5520.bf5 5520k4 $$[0, 1, 0, -116960, -21524892]$$ $$-172798332611391364/94757080078125$$ $$-97031250000000000$$ $$[2]$$ $$43008$$ $$1.9631$$
5520.bf6 5520k6 $$[0, 1, 0, 387120, -95330700]$$ $$3132776881711582558/3735130619961225$$ $$-7649547509680588800$$ $$[4]$$ $$86016$$ $$2.3097$$

## Rank

sage: E.rank()

The elliptic curves in class 5520.bf have rank $$0$$.

## Complex multiplication

The elliptic curves in class 5520.bf do not have complex multiplication.

## Modular form5520.2.a.bf

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.