# Properties

 Label 33600.df Number of curves $8$ Conductor $33600$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("33600.df1")

sage: E.isogeny_class()

## Elliptic curves in class 33600.df

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
33600.df1 33600ex8 [0, -1, 0, -10321633, -8026548863] [2] 2654208
33600.df2 33600ex5 [0, -1, 0, -9217633, -10768452863] [2] 884736
33600.df3 33600ex6 [0, -1, 0, -4321633, 3367451137] [2, 2] 1327104
33600.df4 33600ex3 [0, -1, 0, -4289633, 3421051137] [2] 663552
33600.df5 33600ex2 [0, -1, 0, -577633, -167172863] [2, 2] 442368
33600.df6 33600ex4 [0, -1, 0, -129633, -420292863] [2] 884736
33600.df7 33600ex1 [0, -1, 0, -65633, 2299137] [2] 221184 $$\Gamma_0(N)$$-optimal
33600.df8 33600ex7 [0, -1, 0, 1166367, 11330539137] [2] 2654208

## Rank

sage: E.rank()

The elliptic curves in class 33600.df have rank $$1$$.

## Modular form 33600.2.a.df

sage: E.q_eigenform(10)

$$q - q^{3} + q^{7} + q^{9} + 2q^{13} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.