# Properties

 Label 479808nz Number of curves $4$ Conductor $479808$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("nz1")

sage: E.isogeny_class()

## Elliptic curves in class 479808nz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
479808.nz4 479808nz1 $$[0, 0, 0, 27636, 477812720]$$ $$103823/4386816$$ $$-98629108855140777984$$ $$$$ $$14155776$$ $$2.5154$$ $$\Gamma_0(N)$$-optimal*
479808.nz3 479808nz2 $$[0, 0, 0, -9004044, 10213963760]$$ $$3590714269297/73410624$$ $$1650496493497746456576$$ $$[2, 2]$$ $$28311552$$ $$2.8620$$ $$\Gamma_0(N)$$-optimal*
479808.nz1 479808nz3 $$[0, 0, 0, -143350284, 660610980848]$$ $$14489843500598257/6246072$$ $$140430899131635990528$$ $$$$ $$56623104$$ $$3.2085$$ $$\Gamma_0(N)$$-optimal*
479808.nz2 479808nz4 $$[0, 0, 0, -19164684, -17069386768]$$ $$34623662831857/14438442312$$ $$324620567283633160716288$$ $$$$ $$56623104$$ $$3.2085$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 479808nz1.

## Rank

sage: E.rank()

The elliptic curves in class 479808nz have rank $$0$$.

## Complex multiplication

The elliptic curves in class 479808nz do not have complex multiplication.

## Modular form 479808.2.a.nz

sage: E.q_eigenform(10)

$$q + 2q^{5} - 6q^{13} + q^{17} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 