Properties

 Label 162c Number of curves 4 Conductor 162 CM no Rank 0 Graph Related objects

Show commands for: SageMath
sage: E = EllipticCurve("162.b1")
sage: E.isogeny_class()

Elliptic curves in class 162c

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
162.b4 162c1 [1, -1, 0, 3, -1] 3 6 $$\Gamma_0(N)$$-optimal
162.b3 162c2 [1, -1, 0, -42, -100] 1 18
162.b1 162c3 [1, -1, 0, -1077, 13877] 3 42
162.b2 162c4 [1, -1, 0, -852, 19664] 1 126

Rank

sage: E.rank()

The elliptic curves in class 162c have rank $$0$$.

Modular form162.2.a.b

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

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.

$$\left(\begin{array}{rrrr} 1 & 3 & 7 & 21 \\ 3 & 1 & 21 & 7 \\ 7 & 21 & 1 & 3 \\ 21 & 7 & 3 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels. 