# Properties

 Label 33600.co Number of curves $4$ Conductor $33600$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("co1")

sage: E.isogeny_class()

## Elliptic curves in class 33600.co

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
33600.co1 33600fd4 $$[0, -1, 0, -2229633, -792748863]$$ $$2394165105226952/854262178245$$ $$437382235261440000000$$ $$$$ $$1474560$$ $$2.6615$$
33600.co2 33600fd2 $$[0, -1, 0, -1984633, -1075233863]$$ $$13507798771700416/3544416225$$ $$226842638400000000$$ $$[2, 2]$$ $$737280$$ $$2.3150$$
33600.co3 33600fd1 $$[0, -1, 0, -1984508, -1075376238]$$ $$864335783029582144/59535$$ $$59535000000$$ $$$$ $$368640$$ $$1.9684$$ $$\Gamma_0(N)$$-optimal
33600.co4 33600fd3 $$[0, -1, 0, -1741633, -1348608863]$$ $$-1141100604753992/875529151875$$ $$-448270925760000000000$$ $$$$ $$1474560$$ $$2.6615$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form 33600.2.a.co

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 