Properties

 Label 39270bh Number of curves 8 Conductor 39270 CM no Rank 0 Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("39270.bg1")

sage: E.isogeny_class()

Elliptic curves in class 39270bh

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
39270.bg7 39270bh1 [1, 0, 1, -945654, -2966648] [6] 1327104 $$\Gamma_0(N)$$-optimal
39270.bg5 39270bh2 [1, 0, 1, -10357574, 12789715016] [2, 6] 2654208
39270.bg4 39270bh3 [1, 0, 1, -52843269, -147857449424] [2] 3981312
39270.bg6 39270bh4 [1, 0, 1, -5727074, 24284468216] [6] 5308416
39270.bg2 39270bh5 [1, 0, 1, -165578794, 820064235992] [6] 5308416
39270.bg3 39270bh6 [1, 0, 1, -53846789, -141949927888] [2, 2] 7962624
39270.bg8 39270bh7 [1, 0, 1, 51153211, -628351927888] [2] 15925248
39270.bg1 39270bh8 [1, 0, 1, -174903109, 722537464496] [2] 15925248

Rank

sage: E.rank()

The elliptic curves in class 39270bh have rank $$0$$.

Modular form 39270.2.a.bg

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + 2q^{13} - q^{14} - q^{15} + q^{16} - q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels.