# Properties

 Label 5040bk Number of curves $3$ Conductor $5040$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bk1")

sage: E.isogeny_class()

## Elliptic curves in class 5040bk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5040.v2 5040bk1 $$[0, 0, 0, -192, 1136]$$ $$-262144/35$$ $$-104509440$$ $$[]$$ $$1440$$ $$0.27130$$ $$\Gamma_0(N)$$-optimal
5040.v3 5040bk2 $$[0, 0, 0, 1248, -2896]$$ $$71991296/42875$$ $$-128024064000$$ $$[]$$ $$4320$$ $$0.82061$$
5040.v1 5040bk3 $$[0, 0, 0, -18912, -1047184]$$ $$-250523582464/13671875$$ $$-40824000000000$$ $$[]$$ $$12960$$ $$1.3699$$

## Rank

sage: E.rank()

The elliptic curves in class 5040bk have rank $$0$$.

## Complex multiplication

The elliptic curves in class 5040bk do not have complex multiplication.

## Modular form5040.2.a.bk

sage: E.q_eigenform(10)

$$q + q^{5} - q^{7} - 3q^{11} + 5q^{13} - 3q^{17} - 2q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.