# Properties

 Label 32.a Number of curves $4$ Conductor $32$ CM $$\Q(\sqrt{-1})$$ Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 32.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality CM discriminant
32.a1 32a3 [0, 0, 0, -11, -14] [2] 4   -16
32.a2 32a4 [0, 0, 0, -11, 14] [4] 4   -16
32.a3 32a2 [0, 0, 0, -1, 0] [2, 2] 2   -4
32.a4 32a1 [0, 0, 0, 4, 0] [4] 1 $$\Gamma_0(N)$$-optimal -4

## Rank

sage: E.rank()

The elliptic curves in class 32.a have rank $$0$$.

## Complex multiplication

Each elliptic curve in class 32.a has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-1})$$.

## Modular form32.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{5} - 3q^{9} + 6q^{13} + 2q^{17} + O(q^{20})$$

## 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.