# Properties

 Label 1600p Number of curves 4 Conductor 1600 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("1600.k1")

sage: E.isogeny_class()

## Elliptic curves in class 1600p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1600.k3 1600p1 [0, 0, 0, -200, -1000]  384 $$\Gamma_0(N)$$-optimal
1600.k2 1600p2 [0, 0, 0, -700, 6000] [2, 2] 768
1600.k1 1600p3 [0, 0, 0, -10700, 426000]  1536
1600.k4 1600p4 [0, 0, 0, 1300, 34000]  1536

## Rank

sage: E.rank()

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

## Modular form1600.2.a.k

sage: E.q_eigenform(10)

$$q - 4q^{7} - 3q^{9} + 4q^{11} - 2q^{13} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 