Properties

 Label 6630w Number of curves 8 Conductor 6630 CM no Rank 1 Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("6630.v1")
sage: E.isogeny_class()

Elliptic curves in class 6630w

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
6630.v7 6630w1 [1, 0, 0, -2096146, 1124611076] 12 276480 $$\Gamma_0(N)$$-optimal
6630.v6 6630w2 [1, 0, 0, -5557266, -3547208700] 12 552960
6630.v5 6630w3 [1, 0, 0, -25792786, -50092914940] 4 829440
6630.v4 6630w4 [1, 0, 0, -81373266, -282504599100] 6 1105920
6630.v8 6630w5 [1, 0, 0, 14880814, -23572439484] 6 1105920
6630.v2 6630w6 [1, 0, 0, -411937506, -3218101426764] 4 1658880
6630.v1 6630w7 [1, 0, 0, -6591000006, -205956849489264] 2 3317760
6630.v3 6630w8 [1, 0, 0, -411190526, -3230353542120] 2 3317760

Rank

sage: E.rank()

The elliptic curves in class 6630w have rank $$1$$.

Modular form6630.2.a.v

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

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.