# Properties

 Label 361998.cn Number of curves $6$ Conductor $361998$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("361998.cn1")

sage: E.isogeny_class()

## Elliptic curves in class 361998.cn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
361998.cn1 361998cn5 [1, -1, 1, -20865996716, 1160135576072295] [2] 566231040
361998.cn2 361998cn3 [1, -1, 1, -1306605956, 18054925839591] [2, 2] 283115520
361998.cn3 361998cn6 [1, -1, 1, -445050716, 41508527204967] [2] 566231040
361998.cn4 361998cn2 [1, -1, 1, -137991236, -156765956889] [2, 2] 141557760
361998.cn5 361998cn1 [1, -1, 1, -106841156, -424507124505] [2] 70778880 $$\Gamma_0(N)$$-optimal
361998.cn6 361998cn4 [1, -1, 1, 532222204, -1233396826905] [2] 283115520

## Rank

sage: E.rank()

The elliptic curves in class 361998.cn have rank $$1$$.

## Modular form 361998.2.a.cn

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - 2q^{5} - q^{7} + q^{8} - 2q^{10} + 4q^{11} - q^{14} + q^{16} - q^{17} - 4q^{19} + O(q^{20})$$

## 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 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.