Properties

 Label 198.e Number of curves 4 Conductor 198 CM no Rank 0 Graph Related objects

Show commands for: SageMath
sage: E = EllipticCurve("198.e1")
sage: E.isogeny_class()

Elliptic curves in class 198.e

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
198.e1 198b3 [1, -1, 1, -725, 7661] 6 96
198.e2 198b4 [1, -1, 1, -365, 15005] 6 192
198.e3 198b1 [1, -1, 1, -50, -115] 2 32 $$\Gamma_0(N)$$-optimal
198.e4 198b2 [1, -1, 1, 40, -547] 2 64

Rank

sage: E.rank()

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

Modular form198.2.a.e

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

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 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. 