# Properties

 Label 84966cz Number of curves $6$ Conductor $84966$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("84966.cv1")

sage: E.isogeny_class()

## Elliptic curves in class 84966cz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
84966.cv5 84966cz1 [1, 1, 1, -56939, -11572135] [2] 983040 $$\Gamma_0(N)$$-optimal
84966.cv4 84966cz2 [1, 1, 1, -1189819, -499616839] [2, 2] 1966080
84966.cv3 84966cz3 [1, 1, 1, -1473039, -244152399] [2, 2] 3932160
84966.cv1 84966cz4 [1, 1, 1, -19032679, -31967284735] [2] 3932160
84966.cv6 84966cz5 [1, 1, 1, 5465851, -1878954883] [2] 7864320
84966.cv2 84966cz6 [1, 1, 1, -12943449, 17746038645] [2] 7864320

## Rank

sage: E.rank()

The elliptic curves in class 84966cz have rank $$1$$.

## Modular form 84966.2.a.cv

sage: E.q_eigenform(10)

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

## 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.