Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("198.d1")
sage: E.isogeny_class()

Elliptic curves in class 198.d

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
198.d1 198c4 [1, -1, 1, -1325, 4969] 2 192
198.d2 198c2 [1, -1, 1, -1025, 12881] 6 64
198.d3 198c3 [1, -1, 1, -785, -8207] 2 96
198.d4 198c1 [1, -1, 1, -65, 209] 6 32 $$\Gamma_0(N)$$-optimal

Rank

sage: E.rank()

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

Modular form198.2.a.d

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels. 