# Properties

 Label 33600ez Number of curves 4 Conductor 33600 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("33600.db1")

sage: E.isogeny_class()

## Elliptic curves in class 33600ez

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
33600.db3 33600ez1 [0, -1, 0, -4033, -92063]  49152 $$\Gamma_0(N)$$-optimal
33600.db2 33600ez2 [0, -1, 0, -12033, 395937] [2, 2] 98304
33600.db4 33600ez3 [0, -1, 0, 27967, 2435937]  196608
33600.db1 33600ez4 [0, -1, 0, -180033, 29459937]  196608

## Rank

sage: E.rank()

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

## Modular form 33600.2.a.db

sage: E.q_eigenform(10)

$$q - q^{3} + q^{7} + q^{9} - 6q^{13} - 2q^{17} - 8q^{19} + O(q^{20})$$

## 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. 