# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("ch1")

sage: E.isogeny_class()

## Elliptic curves in class 406560.ch

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
406560.ch1 406560ch4 $$[0, -1, 0, -12349260040, -528209408557400]$$ $$229625675762164624948320008/9568125$$ $$8678664751680000$$ $$[2]$$ $$275251200$$ $$3.9589$$
406560.ch2 406560ch2 $$[0, -1, 0, -774475665, -8193619331775]$$ $$7079962908642659949376/100085966990454375$$ $$726255189063992701048320000$$ $$[2]$$ $$275251200$$ $$3.9589$$ $$\Gamma_0(N)$$-optimal*
406560.ch3 406560ch1 $$[0, -1, 0, -771828790, -8253078202400]$$ $$448487713888272974160064/91549016015625$$ $$10379818647146025000000$$ $$[2, 2]$$ $$137625600$$ $$3.6124$$ $$\Gamma_0(N)$$-optimal*
406560.ch4 406560ch3 $$[0, -1, 0, -769182520, -8312482729868]$$ $$-55486311952875723077768/801237030029296875$$ $$-726753420367734375000000000$$ $$[2]$$ $$275251200$$ $$3.9589$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 406560.ch1.

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 406560.2.a.ch

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 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.