# Properties

 Label 1600o Number of curves $4$ Conductor $1600$ CM $$\Q(\sqrt{-1})$$ Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("o1")

sage: E.isogeny_class()

## Elliptic curves in class 1600o

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality CM discriminant
1600.n4 1600o1 [0, 0, 0, 25, 0] [2] 128 $$\Gamma_0(N)$$-optimal -4
1600.n3 1600o2 [0, 0, 0, -100, 0] [2, 2] 256   -4
1600.n1 1600o3 [0, 0, 0, -1100, -14000] [2] 512   -16
1600.n2 1600o4 [0, 0, 0, -1100, 14000] [2] 512   -16

## Rank

sage: E.rank()

The elliptic curves in class 1600o have rank $$0$$.

## Complex multiplication

Each elliptic curve in class 1600o has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-1})$$.

## Modular form1600.2.a.o

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.