# Properties

 Label 36414.cg Number of curves 6 Conductor 36414 CM no Rank 1 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("36414.cg1")

sage: E.isogeny_class()

## Elliptic curves in class 36414.cg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
36414.cg1 36414cf6 [1, -1, 1, -7102085, 7286730621] [2] 663552
36414.cg2 36414cf5 [1, -1, 1, -443525, 114129789] [2] 331776
36414.cg3 36414cf4 [1, -1, 1, -92390, 8882925] [2] 221184
36414.cg4 36414cf2 [1, -1, 1, -27365, -1734357] [2] 73728
36414.cg5 36414cf1 [1, -1, 1, -1355, -38505] [2] 36864 $$\Gamma_0(N)$$-optimal
36414.cg6 36414cf3 [1, -1, 1, 11650, 809421] [2] 110592

## Rank

sage: E.rank()

The elliptic curves in class 36414.cg have rank $$1$$.

## Modular form 36414.2.a.cg

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{7} + q^{8} - 4q^{13} - q^{14} + q^{16} + 2q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.