# Properties

 Label 428400.ek Number of curves $4$ Conductor $428400$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("ek1")

sage: E.isogeny_class()

## Elliptic curves in class 428400.ek

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
428400.ek1 428400ek3 $$[0, 0, 0, -18284475, 30093430250]$$ $$14489843500598257/6246072$$ $$291416735232000000$$ $$[2]$$ $$18874368$$ $$2.6937$$ $$\Gamma_0(N)$$-optimal*
428400.ek2 428400ek4 $$[0, 0, 0, -2444475, -777577750]$$ $$34623662831857/14438442312$$ $$673639964508672000000$$ $$[2]$$ $$18874368$$ $$2.6937$$
428400.ek3 428400ek2 $$[0, 0, 0, -1148475, 465286250]$$ $$3590714269297/73410624$$ $$3425046073344000000$$ $$[2, 2]$$ $$9437184$$ $$2.3471$$ $$\Gamma_0(N)$$-optimal*
428400.ek4 428400ek1 $$[0, 0, 0, 3525, 21766250]$$ $$103823/4386816$$ $$-204671287296000000$$ $$[2]$$ $$4718592$$ $$2.0006$$ $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 428400.ek1.

## Rank

sage: E.rank()

The elliptic curves in class 428400.ek have rank $$0$$.

## Complex multiplication

The elliptic curves in class 428400.ek do not have complex multiplication.

## Modular form 428400.2.a.ek

sage: E.q_eigenform(10)

$$q - q^{7} + 6q^{13} + q^{17} + O(q^{20})$$

## 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 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.