# Properties

 Label 42.a Number of curves 6 Conductor $42$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("42.a1")
sage: E.isogeny_class()

## Elliptic curves in class 42.a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
42.a1 42a4 [1, 1, 1, -1344, 18405] 4 16
42.a2 42a5 [1, 1, 1, -914, -10915] 2 32
42.a3 42a3 [1, 1, 1, -104, 101] 4 16
42.a4 42a2 [1, 1, 1, -84, 261] 8 8
42.a5 42a1 [1, 1, 1, -4, 5] 8 4 $\Gamma_0(N)$-optimal
42.a6 42a6 [1, 1, 1, 386, 1277] 2 32

## Rank

sage: E.rank()

The elliptic curves in class 42.a have rank $0$.

## Modular form42.2.1.a

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

## Isogeny matrix

sage: E.isogeny_class().matrix()

$\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)$

## Isogeny graph

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