# Properties

 Label 3330f Number of curves $6$ Conductor $3330$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("3330.c1")

sage: E.isogeny_class()

## Elliptic curves in class 3330f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3330.c5 3330f1 [1, -1, 0, -7605, -876875] [2] 13824 $$\Gamma_0(N)$$-optimal
3330.c4 3330f2 [1, -1, 0, -191925, -32248139] [2, 2] 27648
3330.c1 3330f3 [1, -1, 0, -3069045, -2068673675] [2] 55296
3330.c3 3330f4 [1, -1, 0, -263925, -5795339] [2, 2] 55296
3330.c2 3330f5 [1, -1, 0, -2728125, 1727522941] [2] 110592
3330.c6 3330f6 [1, -1, 0, 1048275, -46998419] [2] 110592

## Rank

sage: E.rank()

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

## Modular form3330.2.a.c

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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.