# Properties

 Label 198.b Number of curves 4 Conductor 198 CM no Rank 0 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("198.b1")
sage: E.isogeny_class()

## Elliptic curves in class 198.b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
198.b1 198d4 [1, -1, 0, -9222, -338572] 2 192
198.b2 198d3 [1, -1, 0, -582, -5068] 2 96
198.b3 198d2 [1, -1, 0, -147, -135] 6 64
198.b4 198d1 [1, -1, 0, -87, 333] 6 32 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 198.b have rank $$0$$.

## Modular form198.2.a.b

sage: E.q_eigenform(10)
$$q - q^{2} + q^{4} + 2q^{7} - q^{8} + q^{11} + 2q^{13} - 2q^{14} + q^{16} + 6q^{17} + 2q^{19} + O(q^{20})$$

## 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 LMFDB 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 LMFDB labels.