Properties

 Label 138.b Number of curves 4 Conductor 138 CM no Rank 0 Graph Related objects

Show commands for: SageMath
sage: E = EllipticCurve("138.b1")
sage: E.isogeny_class()

Elliptic curves in class 138.b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
138.b1 138b4 [1, 0, 1, -771, 1342] 2 96
138.b2 138b2 [1, 0, 1, -576, 5266] 6 32
138.b3 138b1 [1, 0, 1, -36, 82] 6 16 $$\Gamma_0(N)$$-optimal
138.b4 138b3 [1, 0, 1, 189, 190] 2 48

Rank

sage: E.rank()

The elliptic curves in class 138.b have rank $$0$$.

Modular form138.2.a.b

sage: E.q_eigenform(10)
$$q - q^{2} + q^{3} + q^{4} - q^{6} + 2q^{7} - q^{8} + q^{9} + q^{12} + 2q^{13} - 2q^{14} + q^{16} - q^{18} + 2q^{19} + O(q^{20})$$

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels. 