# Properties

 Label 1785k Number of curves 4 Conductor 1785 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("1785.c1")

sage: E.isogeny_class()

## Elliptic curves in class 1785k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1785.c4 1785k1 [1, 0, 0, 14, 35]  256 $$\Gamma_0(N)$$-optimal
1785.c3 1785k2 [1, 0, 0, -111, 360] [2, 2] 512
1785.c2 1785k3 [1, 0, 0, -536, -4485]  1024
1785.c1 1785k4 [1, 0, 0, -1686, 26505]  1024

## Rank

sage: E.rank()

The elliptic curves in class 1785k have rank $$1$$.

## Modular form1785.2.a.c

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + q^{7} + 3q^{8} + q^{9} + q^{10} - q^{12} - 6q^{13} - q^{14} - q^{15} - q^{16} - q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## 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. 