# Properties

 Label 330b Number of curves 6 Conductor 330 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("330.e1")

sage: E.isogeny_class()

## Elliptic curves in class 330b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
330.e5 330b1 [1, 0, 0, 5, 17]  32 $$\Gamma_0(N)$$-optimal
330.e4 330b2 [1, 0, 0, -75, 225] [2, 4] 64
330.e3 330b3 [1, 0, 0, -255, -1323] [2, 2] 128
330.e2 330b4 [1, 0, 0, -1175, 15405]  128
330.e1 330b5 [1, 0, 0, -3885, -93525]  256
330.e6 330b6 [1, 0, 0, 495, -7473]  256

## Rank

sage: E.rank()

The elliptic curves in class 330b have rank $$0$$.

## Modular form330.2.a.e

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} - q^{11} + q^{12} - 2q^{13} + q^{15} + q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 