# Properties

 Label 16830bh Number of curves 4 Conductor 16830 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("16830.x1")
sage: E.isogeny_class()

## Elliptic curves in class 16830bh

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
16830.x3 16830bh1 [1, -1, 0, -18594, -2460780] 2 110592 $$\Gamma_0(N)$$-optimal
16830.x2 16830bh2 [1, -1, 0, -410274, -100929132] 2 221184
16830.x4 16830bh3 [1, -1, 0, 162846, 57087828] 6 331776
16830.x1 16830bh4 [1, -1, 0, -1605834, 682139340] 6 663552

## Rank

sage: E.rank()

The elliptic curves in class 16830bh have rank $$1$$.

## Modular form 16830.2.a.x

sage: E.q_eigenform(10)
$$q - q^{2} + q^{4} + q^{5} - 4q^{7} - q^{8} - q^{10} + q^{11} + 2q^{13} + 4q^{14} + q^{16} + q^{17} + 2q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 