# Properties

 Label 411840.i Number of curves $4$ Conductor $411840$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 411840.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
411840.i1 411840i4 $$[0, 0, 0, -923790828, -10807085379152]$$ $$912446049969377120252018/17177299425$$ $$1641316519880294400$$ $$$$ $$88080384$$ $$3.4832$$
411840.i2 411840i3 $$[0, 0, 0, -62886828, -136949964752]$$ $$287849398425814280018/81784533026485575$$ $$7814633826065840101785600$$ $$$$ $$88080384$$ $$3.4832$$ $$\Gamma_0(N)$$-optimal*
411840.i3 411840i2 $$[0, 0, 0, -57738828, -168849031952]$$ $$445574312599094932036/61129333175625$$ $$2920499372689367040000$$ $$[2, 2]$$ $$44040192$$ $$3.1367$$ $$\Gamma_0(N)$$-optimal*
411840.i4 411840i1 $$[0, 0, 0, -3288828, -3125011952]$$ $$-329381898333928144/162600887109375$$ $$-1942094589177600000000$$ $$$$ $$22020096$$ $$2.7901$$ $$\Gamma_0(N)$$-optimal*
*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 411840.i1.

## Rank

sage: E.rank()

The elliptic curves in class 411840.i have rank $$0$$.

## Complex multiplication

The elliptic curves in class 411840.i do not have complex multiplication.

## Modular form 411840.2.a.i

sage: E.q_eigenform(10)

$$q - q^{5} - 4 q^{7} - q^{11} - q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 