# Properties

 Label 3330.d Number of curves $4$ Conductor $3330$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 3330.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3330.d1 3330g3 $$[1, -1, 0, -47475, 3993381]$$ $$16232905099479601/4052240$$ $$2954082960$$ $$[6]$$ $$6912$$ $$1.1928$$
3330.d2 3330g4 $$[1, -1, 0, -47295, 4025025]$$ $$-16048965315233521/256572640900$$ $$-187041455216100$$ $$[6]$$ $$13824$$ $$1.5394$$
3330.d3 3330g1 $$[1, -1, 0, -675, 3861]$$ $$46694890801/18944000$$ $$13810176000$$ $$[2]$$ $$2304$$ $$0.64351$$ $$\Gamma_0(N)$$-optimal
3330.d4 3330g2 $$[1, -1, 0, 2205, 26325]$$ $$1625964918479/1369000000$$ $$-998001000000$$ $$[2]$$ $$4608$$ $$0.99008$$

## Rank

sage: E.rank()

The elliptic curves in class 3330.d have rank $$1$$.

## Complex multiplication

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

## Modular form3330.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{5} + 2q^{7} - q^{8} + q^{10} + 2q^{13} - 2q^{14} + q^{16} - 6q^{17} + 2q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.