Properties

 Label 171b Number of curves 3 Conductor 171 CM no Rank 1 Graph Related objects

Show commands for: SageMath
sage: E = EllipticCurve("171.b1")
sage: E.isogeny_class()

Elliptic curves in class 171b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
171.b3 171b1 [0, 0, 1, 6, 0] 1 8 $$\Gamma_0(N)$$-optimal
171.b2 171b2 [0, 0, 1, -84, 315] 3 24
171.b1 171b3 [0, 0, 1, -6924, 221760] 3 72

Rank

sage: E.rank()

The elliptic curves in class 171b have rank $$1$$.

Modular form171.2.a.b

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels. 