# Properties

 Label 36414bi Number of curves $6$ Conductor $36414$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("36414.k1")

sage: E.isogeny_class()

## Elliptic curves in class 36414bi

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
36414.k5 36414bi1 [1, -1, 0, -10458, -906444] [2] 163840 $$\Gamma_0(N)$$-optimal
36414.k4 36414bi2 [1, -1, 0, -218538, -39234780] [2, 2] 327680
36414.k3 36414bi3 [1, -1, 0, -270558, -19103040] [2, 2] 655360
36414.k1 36414bi4 [1, -1, 0, -3495798, -2514876984] [2] 655360
36414.k6 36414bi5 [1, -1, 0, 1003932, -148336326] [2] 1310720
36414.k2 36414bi6 [1, -1, 0, -2377368, 1397937366] [2] 1310720

## Rank

sage: E.rank()

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

## Modular form 36414.2.a.k

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - 2q^{5} + q^{7} - q^{8} + 2q^{10} - 4q^{11} + 6q^{13} - q^{14} + q^{16} - 4q^{19} + O(q^{20})$$

## 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.