Global number field 40.0.160978304786984968542969308088509683418720896952436305582523345947265625.1

• Label: 40.0.16097830478...5625.1
• Degree: $40$
• Signature: $[0, 20]$
• Discriminant: $5^{70}\cdot 13^{20}$
• Root discriminant: $60.28$
• Ramified primes: $5, 13$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_2\times C_{20}$ (as 40T2)

Normalized defining polynomial

\( x^{40} - 61 x^{35} + 3964 x^{30} - 256627 x^{25} + 16617499 x^{20} + 62360361 x^{15} + 234070236 x^{10} + 875283327 x^{5} + 3486784401 \)

Invariants

Degree:		$40$
Signature:		$[0, 20]$
Discriminant:		\(160978304786984968542969308088509683418720896952436305582523345947265625=5^{70}\cdot 13^{20}\)
Root discriminant:		$60.28$
Ramified primes:		$5, 13$
This field is Galois and abelian over $\Q$.
Conductor:		\(325=5^{2}\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{325}(1,·)$, $\chi_{325}(131,·)$, $\chi_{325}(261,·)$, $\chi_{325}(129,·)$, $\chi_{325}(12,·)$, $\chi_{325}(66,·)$, $\chi_{325}(142,·)$, $\chi_{325}(144,·)$, $\chi_{325}(274,·)$, $\chi_{325}(259,·)$, $\chi_{325}(196,·)$, $\chi_{325}(27,·)$, $\chi_{325}(157,·)$, $\chi_{325}(287,·)$, $\chi_{325}(38,·)$, $\chi_{325}(168,·)$, $\chi_{325}(298,·)$, $\chi_{325}(51,·)$, $\chi_{325}(181,·)$, $\chi_{325}(183,·)$, $\chi_{325}(313,·)$, $\chi_{325}(53,·)$, $\chi_{325}(64,·)$, $\chi_{325}(194,·)$, $\chi_{325}(324,·)$, $\chi_{325}(118,·)$, $\chi_{325}(311,·)$, $\chi_{325}(77,·)$, $\chi_{325}(79,·)$, $\chi_{325}(209,·)$, $\chi_{325}(14,·)$, $\chi_{325}(207,·)$, $\chi_{325}(92,·)$, $\chi_{325}(222,·)$, $\chi_{325}(272,·)$, $\chi_{325}(103,·)$, $\chi_{325}(233,·)$, $\chi_{325}(116,·)$, $\chi_{325}(246,·)$, $\chi_{325}(248,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{19} a^{20} - \frac{2}{19} a^{15} + \frac{4}{19} a^{10} - \frac{8}{19} a^{5} - \frac{3}{19}$, $\frac{1}{57} a^{21} + \frac{17}{57} a^{16} + \frac{4}{57} a^{11} + \frac{11}{57} a^{6} + \frac{16}{57} a$, $\frac{1}{171} a^{22} + \frac{74}{171} a^{17} + \frac{4}{171} a^{12} - \frac{46}{171} a^{7} + \frac{16}{171} a^{2}$, $\frac{1}{513} a^{23} + \frac{74}{513} a^{18} - \frac{167}{513} a^{13} - \frac{46}{513} a^{8} + \frac{187}{513} a^{3}$, $\frac{1}{1539} a^{24} + \frac{587}{1539} a^{19} - \frac{167}{1539} a^{14} + \frac{467}{1539} a^{9} + \frac{187}{1539} a^{4}$, $\frac{1}{76722992883} a^{25} - \frac{64}{4617} a^{20} - \frac{2171}{4617} a^{15} + \frac{1454}{4617} a^{10} - \frac{2186}{4617} a^{5} - \frac{16360872}{315732481}$, $\frac{1}{230168978649} a^{26} - \frac{64}{13851} a^{21} - \frac{6788}{13851} a^{16} - \frac{3163}{13851} a^{11} - \frac{2186}{13851} a^{6} + \frac{299371609}{947197443} a$, $\frac{1}{690506935947} a^{27} - \frac{64}{41553} a^{22} + \frac{7063}{41553} a^{17} + \frac{10688}{41553} a^{12} - \frac{16037}{41553} a^{7} + \frac{1246569052}{2841592329} a^{2}$, $\frac{1}{2071520807841} a^{28} - \frac{64}{124659} a^{23} + \frac{7063}{124659} a^{18} + \frac{52241}{124659} a^{13} + \frac{25516}{124659} a^{8} + \frac{1246569052}{8524776987} a^{3}$, $\frac{1}{6214562423523} a^{29} - \frac{64}{373977} a^{24} + \frac{7063}{373977} a^{19} - \frac{72418}{373977} a^{14} + \frac{150175}{373977} a^{9} - \frac{7278207935}{25574330961} a^{4}$, $\frac{1}{18643687270569} a^{30} - \frac{61}{18643687270569} a^{25} - \frac{14078}{1121931} a^{20} - \frac{514921}{1121931} a^{15} - \frac{59048}{1121931} a^{10} + \frac{425071639}{4038052257} a^{5} - \frac{66466032}{315732481}$, $\frac{1}{55931061811707} a^{31} - \frac{61}{55931061811707} a^{26} - \frac{14078}{3365793} a^{21} - \frac{1636852}{3365793} a^{16} - \frac{1180979}{3365793} a^{11} + \frac{425071639}{12114156771} a^{6} - \frac{382198513}{947197443} a$, $\frac{1}{167793185435121} a^{32} - \frac{61}{167793185435121} a^{27} - \frac{14078}{10097379} a^{22} + \frac{1728941}{10097379} a^{17} + \frac{2184814}{10097379} a^{12} + \frac{425071639}{36342470313} a^{7} - \frac{382198513}{2841592329} a^{2}$, $\frac{1}{503379556305363} a^{33} - \frac{61}{503379556305363} a^{28} - \frac{14078}{30292137} a^{23} + \frac{11826320}{30292137} a^{18} + \frac{2184814}{30292137} a^{13} + \frac{36767541952}{109027410939} a^{8} - \frac{382198513}{8524776987} a^{3}$, $\frac{1}{1510138668916089} a^{34} - \frac{61}{1510138668916089} a^{29} - \frac{14078}{90876411} a^{24} + \frac{11826320}{90876411} a^{19} + \frac{2184814}{90876411} a^{14} + \frac{36767541952}{327082232817} a^{9} - \frac{382198513}{25574330961} a^{4}$, $\frac{1}{4530416006748267} a^{35} - \frac{61}{4530416006748267} a^{30} + \frac{3964}{4530416006748267} a^{25} - \frac{3349273}{272629233} a^{20} + \frac{1}{272629233} a^{15} + \frac{256627}{18643687270569} a^{10} + \frac{3964}{76722992883} a^{5} + \frac{61}{315732481}$, $\frac{1}{13591248020244801} a^{36} - \frac{61}{13591248020244801} a^{31} + \frac{3964}{13591248020244801} a^{26} - \frac{3349273}{817887699} a^{21} + \frac{272629234}{817887699} a^{16} - \frac{18643687013942}{55931061811707} a^{11} + \frac{76722996847}{230168978649} a^{6} - \frac{105244140}{315732481} a$, $\frac{1}{40773744060734403} a^{37} - \frac{61}{40773744060734403} a^{32} + \frac{3964}{40773744060734403} a^{27} - \frac{3349273}{2453663097} a^{22} - \frac{545258465}{2453663097} a^{17} - \frac{74574748825649}{167793185435121} a^{12} + \frac{76722996847}{690506935947} a^{7} + \frac{210488341}{947197443} a^{2}$, $\frac{1}{122321232182203209} a^{38} - \frac{61}{122321232182203209} a^{33} + \frac{3964}{122321232182203209} a^{28} - \frac{3349273}{7360989291} a^{23} - \frac{545258465}{7360989291} a^{18} - \frac{242367934260770}{503379556305363} a^{13} + \frac{767229932794}{2071520807841} a^{8} + \frac{1157685784}{2841592329} a^{3}$, $\frac{1}{366963696546609627} a^{39} - \frac{61}{366963696546609627} a^{34} + \frac{3964}{366963696546609627} a^{29} - \frac{3349273}{22082967873} a^{24} + \frac{6815730826}{22082967873} a^{19} + \frac{261011622044593}{1510138668916089} a^{14} + \frac{2838750740635}{6214562423523} a^{9} + \frac{1157685784}{8524776987} a^{4}$

Class group and class number

Not computed

Unit group

Rank:		$19$
Torsion generator:		\( -\frac{1}{947197443} a^{26} - \frac{727060321}{947197443} a \) (order $50$)
Fundamental units:		Not computed
Regulator:		Not computed

Galois group

$C_2\times C_{20}$ (as 40T2):

An abelian group of order 40

The 40 conjugacy class representatives for $C_2\times C_{20}$

Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{65}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{5}, \sqrt{13})\), 4.0.21125.1, \(\Q(\zeta_{5})\), 5.5.390625.1, 8.0.446265625.1, 10.10.283274078369140625.1, \(\Q(\zeta_{25})^+\), 10.10.56654815673828125.1, 20.20.80244203475886024534702301025390625.1, 20.0.401221017379430122673511505126953125.1, \(\Q(\zeta_{25})\)

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	$20^{2}$	$20^{2}$	R	${\href{/LocalNumberField/7.4.0.1}{4} }^{10}$	${\href{/LocalNumberField/11.10.0.1}{10} }^{4}$	R	$20^{2}$	${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$	$20^{2}$	${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$	$20^{2}$	$20^{2}$	${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
5	Data not computed
13	Data not computed