Properties

 Label 38a Number of curves 3 Conductor 38 CM no Rank 0 Graph Related objects

Show commands for: SageMath
sage: E = EllipticCurve("38.a1")
sage: E.isogeny_class()

Elliptic curves in class 38a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
38.a3 38a1 [1, 0, 1, 9, 90] 3 6 $$\Gamma_0(N)$$-optimal
38.a1 38a2 [1, 0, 1, -86, -2456] 1 18
38.a2 38a3 [1, 0, 1, -16, 22] 3 18

Rank

sage: E.rank()

The elliptic curves in class 38a have rank $$0$$.

Modular form38.2.a.a

sage: E.q_eigenform(10)
$$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} - 2q^{9} - 6q^{11} + q^{12} + 5q^{13} + q^{14} + q^{16} + 3q^{17} + 2q^{18} + q^{19} + O(q^{20})$$

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 Cremona numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels. 