# Properties

 Label 3330.w Number of curves $4$ Conductor $3330$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("w1")

sage: E.isogeny_class()

## Elliptic curves in class 3330.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3330.w1 3330t3 $$[1, -1, 1, -3557, 82531]$$ $$6825481747209/46250$$ $$33716250$$ $$$$ $$2048$$ $$0.62582$$
3330.w2 3330t2 $$[1, -1, 1, -227, 1279]$$ $$1767172329/136900$$ $$99800100$$ $$[2, 2]$$ $$1024$$ $$0.27924$$
3330.w3 3330t1 $$[1, -1, 1, -47, -89]$$ $$15438249/2960$$ $$2157840$$ $$$$ $$512$$ $$-0.067330$$ $$\Gamma_0(N)$$-optimal
3330.w4 3330t4 $$[1, -1, 1, 223, 5419]$$ $$1689410871/18741610$$ $$-13662633690$$ $$$$ $$2048$$ $$0.62582$$

## Rank

sage: E.rank()

The elliptic curves in class 3330.w have rank $$0$$.

## Complex multiplication

The elliptic curves in class 3330.w do not have complex multiplication.

## Modular form3330.2.a.w

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} + 4q^{11} + 2q^{13} + q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$ ## 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 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. 