Number field 39.39.118129027602237903625231846226768806470030318124152946028448455403348348206499160462586488797476753449.1

• Label: 39.39.118...449.1
• Degree: $39$
• Signature: $[39, 0]$
• Discriminant: $1.181\times 10^{101}$
• Root discriminant: \(390.48\)
• Ramified primes: $13,79$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{39}$ (as 39T1)

Normalized defining polynomial

x^39 - x^38 - 354*x^37 + 511*x^36 + 56045*x^35 - 103251*x^34 - 5247131*x^33 + 11548919*x^32 + 323858271*x^31 - 818077472*x^30 - 13915010959*x^29 + 39315623658*x^28 + 428683737326*x^27 - 1332897877353*x^26 - 9615971778232*x^25 + 32610615529383*x^24 + 157954520735238*x^23 - 583064031850378*x^22 - 1895313576585435*x^21 + 7659727015753552*x^20 + 16422194396956823*x^19 - 73899082723108593*x^18 - 100210603740569843*x^17 + 520414495467180616*x^16 + 408676600606594002*x^15 - 2642936340868695765*x^14 - 971974365656650043*x^13 + 9496075566431610394*x^12 + 586468002053107902*x^11 - 23471590571480299925*x^10 + 3789203055222994557*x^9 + 38292325977191105354*x^8 - 12389487729343041691*x^7 - 38551672930076472707*x^6 + 17045767379408771012*x^5 + 21043564296778861782*x^4 - 11248804489468062285*x^3 - 4368569542269920485*x^2 + 2837239422181036677*x - 221647485396299581

Invariants

Degree:		$39$
Signature:		$[39, 0]$
Discriminant:		118129027602237903625231846226768806470030318124152946028448455403348348206499160462586488797476753449
Root discriminant:		\(390.48\)
Galois root discriminant:		$13^{2/3}79^{38/39}\approx 390.4800838648001$
Ramified primes:		\(13\), \(79\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$39$
This field is Galois and abelian over $\Q$.
Conductor:		\(1027=13\cdot 79\)
Dirichlet character group:		$\lbrace$$\chi_{1027}(256,·)$, $\chi_{1027}(1,·)$, $\chi_{1027}(131,·)$, $\chi_{1027}(9,·)$, $\chi_{1027}(523,·)$, $\chi_{1027}(269,·)$, $\chi_{1027}(399,·)$, $\chi_{1027}(16,·)$, $\chi_{1027}(913,·)$, $\chi_{1027}(919,·)$, $\chi_{1027}(152,·)$, $\chi_{1027}(367,·)$, $\chi_{1027}(672,·)$, $\chi_{1027}(417,·)$, $\chi_{1027}(347,·)$, $\chi_{1027}(42,·)$, $\chi_{1027}(55,·)$, $\chi_{1027}(321,·)$, $\chi_{1027}(835,·)$, $\chi_{1027}(196,·)$, $\chi_{1027}(326,·)$, $\chi_{1027}(971,·)$, $\chi_{1027}(81,·)$, $\chi_{1027}(341,·)$, $\chi_{1027}(599,·)$, $\chi_{1027}(729,·)$, $\chi_{1027}(731,·)$, $\chi_{1027}(378,·)$, $\chi_{1027}(222,·)$, $\chi_{1027}(471,·)$, $\chi_{1027}(737,·)$, $\chi_{1027}(482,·)$, $\chi_{1027}(230,·)$, $\chi_{1027}(144,·)$, $\chi_{1027}(495,·)$, $\chi_{1027}(880,·)$, $\chi_{1027}(1015,·)$, $\chi_{1027}(250,·)$, $\chi_{1027}(510,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{157}a^{36}+\frac{26}{157}a^{35}-\frac{2}{157}a^{34}-\frac{24}{157}a^{33}-\frac{54}{157}a^{32}-\frac{36}{157}a^{31}+\frac{66}{157}a^{30}+\frac{16}{157}a^{29}-\frac{28}{157}a^{28}+\frac{40}{157}a^{27}-\frac{60}{157}a^{26}+\frac{70}{157}a^{25}+\frac{34}{157}a^{24}-\frac{38}{157}a^{23}-\frac{46}{157}a^{22}+\frac{13}{157}a^{21}-\frac{58}{157}a^{20}-\frac{57}{157}a^{19}-\frac{45}{157}a^{18}-\frac{62}{157}a^{17}+\frac{57}{157}a^{16}+\frac{30}{157}a^{15}-\frac{37}{157}a^{14}+\frac{65}{157}a^{13}-\frac{65}{157}a^{12}+\frac{45}{157}a^{11}+\frac{6}{157}a^{10}+\frac{10}{157}a^{9}+\frac{62}{157}a^{8}+\frac{15}{157}a^{7}-\frac{13}{157}a^{6}-\frac{51}{157}a^{5}+\frac{57}{157}a^{4}-\frac{32}{157}a^{3}-\frac{66}{157}a^{2}-\frac{64}{157}a+\frac{22}{157}$, $\frac{1}{19274419}a^{37}+\frac{44507}{19274419}a^{36}+\frac{3136588}{19274419}a^{35}+\frac{3112715}{19274419}a^{34}-\frac{8344548}{19274419}a^{33}+\frac{7862807}{19274419}a^{32}+\frac{7599421}{19274419}a^{31}+\frac{9412012}{19274419}a^{30}-\frac{6847097}{19274419}a^{29}-\frac{7871770}{19274419}a^{28}+\frac{8024797}{19274419}a^{27}+\frac{4137631}{19274419}a^{26}-\frac{9558551}{19274419}a^{25}+\frac{4906499}{19274419}a^{24}-\frac{3666483}{19274419}a^{23}-\frac{6222313}{19274419}a^{22}-\frac{4108412}{19274419}a^{21}+\frac{4970175}{19274419}a^{20}-\frac{9398246}{19274419}a^{19}-\frac{3195221}{19274419}a^{18}-\frac{433066}{19274419}a^{17}+\frac{5066915}{19274419}a^{16}-\frac{500309}{19274419}a^{15}-\frac{6092600}{19274419}a^{14}+\frac{9440141}{19274419}a^{13}-\frac{1756581}{19274419}a^{12}+\frac{5679062}{19274419}a^{11}-\frac{7645119}{19274419}a^{10}+\frac{4475062}{19274419}a^{9}+\frac{10758}{46001}a^{8}+\frac{8580002}{19274419}a^{7}-\frac{2512544}{19274419}a^{6}+\frac{8026173}{19274419}a^{5}+\frac{6185007}{19274419}a^{4}+\frac{5479675}{19274419}a^{3}-\frac{444377}{19274419}a^{2}+\frac{2570209}{19274419}a-\frac{7086979}{19274419}$, $\frac{1}{34\!\cdots\!13}a^{38}-\frac{56\!\cdots\!30}{34\!\cdots\!13}a^{37}-\frac{84\!\cdots\!55}{34\!\cdots\!13}a^{36}-\frac{13\!\cdots\!10}{34\!\cdots\!13}a^{35}+\frac{15\!\cdots\!06}{34\!\cdots\!13}a^{34}-\frac{91\!\cdots\!26}{34\!\cdots\!13}a^{33}-\frac{11\!\cdots\!98}{34\!\cdots\!13}a^{32}+\frac{80\!\cdots\!71}{34\!\cdots\!13}a^{31}+\frac{13\!\cdots\!53}{34\!\cdots\!13}a^{30}+\frac{45\!\cdots\!80}{34\!\cdots\!13}a^{29}+\frac{21\!\cdots\!17}{34\!\cdots\!13}a^{28}+\frac{16\!\cdots\!09}{34\!\cdots\!13}a^{27}-\frac{10\!\cdots\!70}{34\!\cdots\!13}a^{26}+\frac{11\!\cdots\!55}{34\!\cdots\!13}a^{25}+\frac{14\!\cdots\!83}{34\!\cdots\!13}a^{24}+\frac{10\!\cdots\!67}{34\!\cdots\!13}a^{23}+\frac{70\!\cdots\!87}{34\!\cdots\!13}a^{22}-\frac{15\!\cdots\!25}{34\!\cdots\!13}a^{21}+\frac{61\!\cdots\!10}{34\!\cdots\!13}a^{20}+\frac{14\!\cdots\!46}{34\!\cdots\!13}a^{19}+\frac{87\!\cdots\!72}{34\!\cdots\!13}a^{18}+\frac{15\!\cdots\!78}{34\!\cdots\!13}a^{17}+\frac{96\!\cdots\!46}{34\!\cdots\!13}a^{16}-\frac{13\!\cdots\!31}{34\!\cdots\!13}a^{15}-\frac{12\!\cdots\!61}{34\!\cdots\!13}a^{14}-\frac{57\!\cdots\!37}{34\!\cdots\!13}a^{13}+\frac{77\!\cdots\!42}{34\!\cdots\!13}a^{12}+\frac{15\!\cdots\!64}{34\!\cdots\!13}a^{11}+\frac{92\!\cdots\!65}{34\!\cdots\!13}a^{10}-\frac{21\!\cdots\!32}{34\!\cdots\!13}a^{9}+\frac{12\!\cdots\!34}{34\!\cdots\!13}a^{8}+\frac{98\!\cdots\!68}{34\!\cdots\!13}a^{7}-\frac{12\!\cdots\!64}{34\!\cdots\!13}a^{6}+\frac{18\!\cdots\!32}{34\!\cdots\!13}a^{5}+\frac{11\!\cdots\!48}{34\!\cdots\!13}a^{4}+\frac{17\!\cdots\!96}{34\!\cdots\!13}a^{3}-\frac{15\!\cdots\!14}{34\!\cdots\!13}a^{2}+\frac{11\!\cdots\!61}{34\!\cdots\!13}a+\frac{45\!\cdots\!32}{34\!\cdots\!13}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

Unit group

Rank:		$38$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

Galois group

$C_{39}$ (as 39T1):

A cyclic group of order 39

The 39 conjugacy class representatives for $C_{39}$

Character table for $C_{39}$ is not computed

Intermediate fields

3.3.1054729.1, 13.13.59091511031674153381441.1

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	${\href{/padicField/2.13.0.1}{13} }^{3}$	${\href{/padicField/3.13.0.1}{13} }^{3}$	$39$	${\href{/padicField/7.13.0.1}{13} }^{3}$	$39$	R	$39$	${\href{/padicField/19.13.0.1}{13} }^{3}$	${\href{/padicField/23.3.0.1}{3} }^{13}$	$39$	$39$	${\href{/padicField/37.13.0.1}{13} }^{3}$	$39$	$39$	$39$	$39$	$39$

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
\(13\)		Deg $39$	$3$	$13$	$26$
\(79\)		Deg $39$	$39$	$1$	$38$