# Properties

 Label 406560bs Number of curves $4$ Conductor $406560$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bs1")

sage: E.isogeny_class()

## Elliptic curves in class 406560bs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
406560.bs3 406560bs1 $$[0, -1, 0, -8510, 300192]$$ $$601211584/11025$$ $$1250013441600$$ $$[2, 2]$$ $$983040$$ $$1.1160$$ $$\Gamma_0(N)$$-optimal*
406560.bs1 406560bs2 $$[0, -1, 0, -135560, 19256052]$$ $$303735479048/105$$ $$95239119360$$ $$[2]$$ $$1966080$$ $$1.4625$$ $$\Gamma_0(N)$$-optimal*
406560.bs4 406560bs3 $$[0, -1, 0, -40, 862600]$$ $$-8/354375$$ $$-321432027840000$$ $$[2]$$ $$1966080$$ $$1.4625$$
406560.bs2 406560bs4 $$[0, -1, 0, -17585, -442143]$$ $$82881856/36015$$ $$261336143523840$$ $$[2]$$ $$1966080$$ $$1.4625$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 406560bs1.

## Rank

sage: E.rank()

The elliptic curves in class 406560bs have rank $$1$$.

## Complex multiplication

The elliptic curves in class 406560bs do not have complex multiplication.

## Modular form 406560.2.a.bs

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} - q^{7} + q^{9} - 6q^{13} - q^{15} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.