Global number field 40.40.130305099804548492884220428175380349368393046678311823693003457545895936.1: \(\Q(\zeta_{264})^+\)

• Label: 40.40.1303050998...5936.1
• Degree: $40$
• Signature: $[40, 0]$
• Discriminant: $2^{80}\cdot 3^{20}\cdot 11^{36}$
• Root discriminant: $59.96$
• Ramified primes: $2, 3, 11$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_2^2\times C_{10}$ (as 40T7)

Normalized defining polynomial

\( x^{40} - 40 x^{38} + 741 x^{36} - 8436 x^{34} + 66044 x^{32} - 376960 x^{30} + 1622694 x^{28} - 5375528 x^{26} + 13860054 x^{24} - 27947920 x^{22} + 44043506 x^{20} - 53927016 x^{18} + 50713585 x^{16} - 35964944 x^{14} + 18713229 x^{12} - 6857500 x^{10} + 1662386 x^{8} - 240976 x^{6} + 17560 x^{4} - 480 x^{2} + 1 \)

Invariants

Degree:		$40$
Signature:		$[40, 0]$
Discriminant:		\(130305099804548492884220428175380349368393046678311823693003457545895936=2^{80}\cdot 3^{20}\cdot 11^{36}\)
Root discriminant:		$59.96$
Ramified primes:		$2, 3, 11$
This field is Galois and abelian over $\Q$.
Conductor:		\(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(259,·)$, $\chi_{264}(133,·)$, $\chi_{264}(7,·)$, $\chi_{264}(139,·)$, $\chi_{264}(175,·)$, $\chi_{264}(17,·)$, $\chi_{264}(19,·)$, $\chi_{264}(149,·)$, $\chi_{264}(23,·)$, $\chi_{264}(25,·)$, $\chi_{264}(155,·)$, $\chi_{264}(29,·)$, $\chi_{264}(161,·)$, $\chi_{264}(37,·)$, $\chi_{264}(41,·)$, $\chi_{264}(43,·)$, $\chi_{264}(173,·)$, $\chi_{264}(47,·)$, $\chi_{264}(49,·)$, $\chi_{264}(179,·)$, $\chi_{264}(181,·)$, $\chi_{264}(59,·)$, $\chi_{264}(151,·)$, $\chi_{264}(191,·)$, $\chi_{264}(65,·)$, $\chi_{264}(197,·)$, $\chi_{264}(71,·)$, $\chi_{264}(203,·)$, $\chi_{264}(79,·)$, $\chi_{264}(211,·)$, $\chi_{264}(169,·)$, $\chi_{264}(101,·)$, $\chi_{264}(97,·)$, $\chi_{264}(229,·)$, $\chi_{264}(233,·)$, $\chi_{264}(157,·)$, $\chi_{264}(119,·)$, $\chi_{264}(251,·)$, $\chi_{264}(127,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$39$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 73555688392588420000000 \) (assuming GRH)

Galois group

$C_2^2\times C_{10}$ (as 40T7):

An abelian group of order 40

The 40 conjugacy class representatives for $C_2^2\times C_{10}$

Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{22}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{66}) \), \(\Q(\sqrt{6}, \sqrt{22})\), \(\Q(\sqrt{2}, \sqrt{11})\), \(\Q(\sqrt{3}, \sqrt{22})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{6}, \sqrt{11})\), \(\Q(\sqrt{2}, \sqrt{33})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 8.8.77720518656.1, 10.10.77265229938688.1, 10.10.1706859170463744.1, \(\Q(\zeta_{33})^+\), 10.10.7024111812608.1, \(\Q(\zeta_{44})^+\), 10.10.53339349076992.1, 10.10.18775450875101184.1, 20.20.352517555563337816067682238201856.2, \(\Q(\zeta_{88})^+\), 20.20.360977976896857923653306611918700544.1, 20.20.2983289065263288625233938941476864.1, 20.20.360977976896857923653306611918700544.2, 20.20.352517555563337816067682238201856.1, \(\Q(\zeta_{132})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	R	${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/7.10.0.1}{10} }^{4}$	R	${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$	${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/37.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$	${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
2	Data not computed
3	Data not computed
11	Data not computed