# Properties

 Label 32368.q Number of curves $4$ Conductor $32368$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("q1")

sage: E.isogeny_class()

## Elliptic curves in class 32368.q

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
32368.q1 32368a4 [0, 0, 0, -86411, -9776870] [2] 73728
32368.q2 32368a3 [0, 0, 0, -17051, 677994] [2] 73728
32368.q3 32368a2 [0, 0, 0, -5491, -147390] [2, 2] 36864
32368.q4 32368a1 [0, 0, 0, 289, -9826] [2] 18432 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 32368.q have rank $$1$$.

## Complex multiplication

The elliptic curves in class 32368.q do not have complex multiplication.

## Modular form 32368.2.a.q

sage: E.q_eigenform(10)

$$q - 2q^{5} - q^{7} - 3q^{9} - 4q^{11} + 2q^{13} - 8q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.