# Properties

 Label 33600gi Number of curves $6$ Conductor $33600$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("33600.dz1")

sage: E.isogeny_class()

## Elliptic curves in class 33600gi

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
33600.dz5 33600gi1 [0, 1, 0, -6433, -440737] [2] 98304 $$\Gamma_0(N)$$-optimal
33600.dz4 33600gi2 [0, 1, 0, -134433, -19000737] [2, 2] 196608
33600.dz3 33600gi3 [0, 1, 0, -166433, -9304737] [2, 2] 393216
33600.dz1 33600gi4 [0, 1, 0, -2150433, -1214488737] [2] 393216
33600.dz6 33600gi5 [0, 1, 0, 617567, -71240737] [2] 786432
33600.dz2 33600gi6 [0, 1, 0, -1462433, 673687263] [2] 786432

## Rank

sage: E.rank()

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

## Modular form 33600.2.a.dz

sage: E.q_eigenform(10)

$$q + q^{3} - q^{7} + q^{9} - 4q^{11} + 6q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.