# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("16830.t1")
sage: E.isogeny_class()

## Elliptic curves in class 16830z

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
16830.t4 16830z1 [1, -1, 0, -1494, -123660] 2 40960 $$\Gamma_0(N)$$-optimal
16830.t3 16830z2 [1, -1, 0, -47574, -3966732] 4 81920
16830.t1 16830z3 [1, -1, 0, -760374, -255014892] 2 163840
16830.t2 16830z4 [1, -1, 0, -72054, 571860] 2 163840

## Rank

sage: E.rank()

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

## Modular form 16830.2.a.t

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

## 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. 