# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("84966.cr1")

sage: E.isogeny_class()

## Elliptic curves in class 84966.cr

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
84966.cr1 84966da6 [1, 1, 1, -194269151539, 32957388273621545] [2] 424673280
84966.cr2 84966da4 [1, 1, 1, -12164922379, 512897547867641] [2, 2] 212336640
84966.cr3 84966da5 [1, 1, 1, -4143565539, 1179186740968137] [2] 424673280
84966.cr4 84966da2 [1, 1, 1, -1284742859, -4454988308359] [2, 2] 106168320
84966.cr5 84966da1 [1, 1, 1, -994725579, -12060749479815] [2] 53084160 $$\Gamma_0(N)$$-optimal
84966.cr6 84966da3 [1, 1, 1, 4955160181, -35033009165575] [2] 212336640

## Rank

sage: E.rank()

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

## Modular form 84966.2.a.cr

sage: E.q_eigenform(10)

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

## 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 LMFDB 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 LMFDB labels.