Properties

 Label 6270l Number of curves 8 Conductor 6270 CM no Rank 0 Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("6270.l1")
sage: E.isogeny_class()

Elliptic curves in class 6270l

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
6270.l7 6270l1 [1, 0, 1, -86508, 8587306] 6 64512 $$\Gamma_0(N)$$-optimal
6270.l5 6270l2 [1, 0, 1, -1336508, 594587306] 12 129024
6270.l4 6270l3 [1, 0, 1, -1745883, -886935194] 2 193536
6270.l2 6270l4 [1, 0, 1, -21384008, 38059355306] 6 258048
6270.l6 6270l5 [1, 0, 1, -1289008, 638819306] 6 258048
6270.l3 6270l6 [1, 0, 1, -2257883, -324349594] 4 387072
6270.l1 6270l7 [1, 0, 1, -21424283, 37908785126] 2 774144
6270.l8 6270l8 [1, 0, 1, 8716517, -2549957914] 2 774144

Rank

sage: E.rank()

The elliptic curves in class 6270l have rank $$0$$.

Modular form6270.2.a.l

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.