Ranks of elliptic curves in cyclic sextic extensions of $\mathbb{Q}$

Abstract: For an elliptic curve

E / m a t h b b Q

we show that there are infinitely many cyclic sextic extensions

K / m a t h b b Q

such that the Mordell-Weil group

E (K)

has rank greater than the subgroup of

E (K)

generated by all the

E (F)

for the proper subfields

F s u b s e t K

. For certain curves

E / m a t h b b Q

we show that the number of such fields

K

of conductor less than

X

is

g g s q r t X

.

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