Properties

 Label 27a Number of curves 4 Conductor 27 CM -3 Rank 0 Graph Related objects

Show commands for: SageMath
sage: E = EllipticCurve("27.a1")
sage: E.isogeny_class()

Elliptic curves in class 27a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
27.a3 27a1 [0, 0, 1, 0, -7] 3 1 $$\Gamma_0(N)$$-optimal
27.a1 27a2 [0, 0, 1, -270, -1708] 1 3
27.a4 27a3 [0, 0, 1, 0, 0] 3 3
27.a2 27a4 [0, 0, 1, -30, 63] 3 9

Rank

sage: E.rank()

The elliptic curves in class 27a have rank $$0$$.

Modular form27.2.a.a

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

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.

$$\left(\begin{array}{rrrr} 1 & 3 & 3 & 9 \\ 3 & 1 & 9 & 27 \\ 3 & 9 & 1 & 3 \\ 9 & 27 & 3 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels. 