# Properties

 Label 882e Number of curves $6$ Conductor $882$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("882.b1")

sage: E.isogeny_class()

## Elliptic curves in class 882e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
882.b5 882e1 [1, -1, 0, -1773, 63909] [2] 1536 $$\Gamma_0(N)$$-optimal
882.b4 882e2 [1, -1, 0, -37053, 2752245] [2, 2] 3072
882.b3 882e3 [1, -1, 0, -45873, 1349865] [2, 2] 6144
882.b1 882e4 [1, -1, 0, -592713, 175784769] [2] 6144
882.b2 882e5 [1, -1, 0, -403083, -97454421] [2] 12288
882.b6 882e6 [1, -1, 0, 170217, 10295991] [2] 12288

## Rank

sage: E.rank()

The elliptic curves in class 882e have rank $$1$$.

## Modular form882.2.a.b

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.