# Properties

 Label 194271.w Number of curves 6 Conductor 194271 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("194271.w1")

sage: E.isogeny_class()

## Elliptic curves in class 194271.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
194271.w1 194271u6 [1, 0, 1, -3800497, -2852046799] [2] 4014080
194271.w2 194271u4 [1, 0, 1, -238862, -44053765] [2, 2] 2007040
194271.w3 194271u2 [1, 0, 1, -32817, 1276135] [2, 2] 1003520
194271.w4 194271u1 [1, 0, 1, -28612, 1859789] [2] 501760 $$\Gamma_0(N)$$-optimal
194271.w5 194271u5 [1, 0, 1, 26053, -136350151] [2] 4014080
194271.w6 194271u3 [1, 0, 1, 105948, 9268999] [2] 2007040

## Rank

sage: E.rank()

The elliptic curves in class 194271.w have rank $$0$$.

## Modular form 194271.2.a.w

sage: E.q_eigenform(10)

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

## Isogeny matrix

sage: E.isogeny_class().matrix()

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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.