# Properties

 Label 406560fg Number of curves $4$ Conductor $406560$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("fg1")

sage: E.isogeny_class()

## Elliptic curves in class 406560fg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
406560.fg3 406560fg1 $$[0, 1, 0, -8510, -300192]$$ $$601211584/11025$$ $$1250013441600$$ $$[2, 2]$$ $$983040$$ $$1.1160$$ $$\Gamma_0(N)$$-optimal*
406560.fg2 406560fg2 $$[0, 1, 0, -17585, 442143]$$ $$82881856/36015$$ $$261336143523840$$ $$$$ $$1966080$$ $$1.4625$$ $$\Gamma_0(N)$$-optimal*
406560.fg4 406560fg3 $$[0, 1, 0, -40, -862600]$$ $$-8/354375$$ $$-321432027840000$$ $$$$ $$1966080$$ $$1.4625$$
406560.fg1 406560fg4 $$[0, 1, 0, -135560, -19256052]$$ $$303735479048/105$$ $$95239119360$$ $$$$ $$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 406560fg1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 406560.2.a.fg

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. 