# Properties

 Label 33600en Number of curves $4$ Conductor $33600$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("en1")

sage: E.isogeny_class()

## Elliptic curves in class 33600en

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
33600.ch3 33600en1 $$[0, -1, 0, -908, 5562]$$ $$82881856/36015$$ $$36015000000$$ $$[2]$$ $$36864$$ $$0.72173$$ $$\Gamma_0(N)$$-optimal
33600.ch2 33600en2 $$[0, -1, 0, -7033, -221063]$$ $$601211584/11025$$ $$705600000000$$ $$[2, 2]$$ $$73728$$ $$1.0683$$
33600.ch4 33600en3 $$[0, -1, 0, -33, -648063]$$ $$-8/354375$$ $$-181440000000000$$ $$[2]$$ $$147456$$ $$1.4149$$
33600.ch1 33600en4 $$[0, -1, 0, -112033, -14396063]$$ $$303735479048/105$$ $$53760000000$$ $$[2]$$ $$147456$$ $$1.4149$$

## Rank

sage: E.rank()

The elliptic curves in class 33600en have rank $$0$$.

## Complex multiplication

The elliptic curves in class 33600en do not have complex multiplication.

## Modular form 33600.2.a.en

sage: E.q_eigenform(10)

$$q - q^{3} - q^{7} + q^{9} + 4q^{11} + 6q^{13} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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 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.