# Properties

 Label 1785.b Number of curves 4 Conductor 1785 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("1785.b1")

sage: E.isogeny_class()

## Elliptic curves in class 1785.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1785.b1 1785h3 [1, 1, 1, -1015, 9182]  1536
1785.b2 1785h2 [1, 1, 1, -340, -2428] [2, 2] 768
1785.b3 1785h1 [1, 1, 1, -335, -2500]  384 $$\Gamma_0(N)$$-optimal
1785.b4 1785h4 [1, 1, 1, 255, -9330]  1536

## Rank

sage: E.rank()

The elliptic curves in class 1785.b have rank $$0$$.

## Modular form1785.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + q^{7} + 3q^{8} + q^{9} - q^{10} + 4q^{11} + q^{12} + 2q^{13} - q^{14} - q^{15} - q^{16} + q^{17} - q^{18} + 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 