# Properties

 Label 321552er Number of curves $4$ Conductor $321552$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("er1")

sage: E.isogeny_class()

## Elliptic curves in class 321552er

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
321552.er4 321552er1 $$[0, 0, 0, 84141, 10707282]$$ $$22062729659823/29354283343$$ $$-87651420393664512$$ $$[2]$$ $$2752512$$ $$1.9368$$ $$\Gamma_0(N)$$-optimal
321552.er3 321552er2 $$[0, 0, 0, -521379, 104805090]$$ $$5249244962308257/1448621666569$$ $$4325561118428368896$$ $$[2, 2]$$ $$5505024$$ $$2.2834$$
321552.er1 321552er3 $$[0, 0, 0, -7683219, 8196251922]$$ $$16798320881842096017/2132227789307$$ $$6366798063226073088$$ $$[2]$$ $$11010048$$ $$2.6300$$
321552.er2 321552er4 $$[0, 0, 0, -3047859, -1964382030]$$ $$1048626554636928177/48569076788309$$ $$145026486184662061056$$ $$[2]$$ $$11010048$$ $$2.6300$$

## Rank

sage: E.rank()

The elliptic curves in class 321552er have rank $$0$$.

## Complex multiplication

The elliptic curves in class 321552er do not have complex multiplication.

## Modular form 321552.2.a.er

sage: E.q_eigenform(10)

$$q + 2q^{5} + q^{7} - q^{11} + 6q^{13} + 2q^{17} + 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.