Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("162.c1")
sage: E.isogeny_class()

Elliptic curves in class 162b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
162.c3 162b1 [1, -1, 1, -5, 5] 3 6 $$\Gamma_0(N)$$-optimal
162.c4 162b2 [1, -1, 1, 25, 1] 1 18
162.c2 162b3 [1, -1, 1, -95, -697] 3 42
162.c1 162b4 [1, -1, 1, -9695, -364985] 1 126

Rank

sage: E.rank()

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

Modular form162.2.a.c

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. 