Number field 39.39.1287743804278744050410620426954739687963064854495168753870500853746064161.1: \(\Q(\zeta_{79})^+\)

• Label: 39.39.128...161.1
• Degree: $39$
• Signature: $[39, 0]$
• Discriminant: $1.288\times 10^{72}$
• Root discriminant: \(70.63\)
• Ramified prime: $79$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{39}$ (as 39T1)

Normalized defining polynomial

x^39 - x^38 - 38*x^37 + 37*x^36 + 666*x^35 - 630*x^34 - 7140*x^33 + 6545*x^32 + 52360*x^31 - 46376*x^30 - 278256*x^29 + 237336*x^28 + 1107568*x^27 - 906192*x^26 - 3365856*x^25 + 2629575*x^24 + 7888725*x^23 - 5852925*x^22 - 14307150*x^21 + 10015005*x^20 + 20030010*x^19 - 13123110*x^18 - 21474180*x^17 + 13037895*x^16 + 17383860*x^15 - 9657700*x^14 - 10400600*x^13 + 5200300*x^12 + 4457400*x^11 - 1961256*x^10 - 1307504*x^9 + 490314*x^8 + 245157*x^7 - 74613*x^6 - 26334*x^5 + 5985*x^4 + 1330*x^3 - 190*x^2 - 20*x + 1

Invariants

Degree:		$39$
Signature:		$[39, 0]$
Discriminant:		\(1287743804278744050410620426954739687963064854495168753870500853746064161\) \(\medspace = 79^{38}\)
Root discriminant:		\(70.63\)
Galois root discriminant:		$79^{38/39}\approx 70.626874311955$
Ramified primes:		\(79\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$39$
This field is Galois and abelian over $\Q$.
Conductor:		\(79\)
Dirichlet character group:		$\lbrace$$\chi_{79}(1,·)$, $\chi_{79}(2,·)$, $\chi_{79}(4,·)$, $\chi_{79}(5,·)$, $\chi_{79}(8,·)$, $\chi_{79}(9,·)$, $\chi_{79}(10,·)$, $\chi_{79}(11,·)$, $\chi_{79}(13,·)$, $\chi_{79}(16,·)$, $\chi_{79}(18,·)$, $\chi_{79}(19,·)$, $\chi_{79}(20,·)$, $\chi_{79}(21,·)$, $\chi_{79}(22,·)$, $\chi_{79}(23,·)$, $\chi_{79}(25,·)$, $\chi_{79}(26,·)$, $\chi_{79}(31,·)$, $\chi_{79}(32,·)$, $\chi_{79}(36,·)$, $\chi_{79}(38,·)$, $\chi_{79}(40,·)$, $\chi_{79}(42,·)$, $\chi_{79}(44,·)$, $\chi_{79}(45,·)$, $\chi_{79}(46,·)$, $\chi_{79}(49,·)$, $\chi_{79}(50,·)$, $\chi_{79}(51,·)$, $\chi_{79}(52,·)$, $\chi_{79}(55,·)$, $\chi_{79}(62,·)$, $\chi_{79}(64,·)$, $\chi_{79}(65,·)$, $\chi_{79}(67,·)$, $\chi_{79}(72,·)$, $\chi_{79}(73,·)$, $\chi_{79}(76,·)$$\rbrace$
This is not a CM field.

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

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

Monogenic:		Yes
Index:		$1$
Inessential primes:		None

Class group and class number

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

Unit group

Rank:		$38$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+2$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314610a^{9}-286824a^{7}+35853a^{5}-2109a^{3}+37a-1$, $a^{4}-4a^{2}+2$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}-a^{24}+20150a^{23}+24a^{22}-78430a^{21}-252a^{20}+219604a^{19}+1520a^{18}-447050a^{17}-5814a^{16}+660841a^{15}+14688a^{14}-700791a^{13}-24752a^{12}+520234a^{11}+27456a^{10}-259403a^{9}-19305a^{8}+81091a^{7}+8008a^{6}-14049a^{5}-1716a^{4}+1050a^{3}+144a^{2}-21a-3$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{34}-34a^{32}+527a^{30}-4930a^{28}+31059a^{26}-139230a^{24}-a^{23}+457471a^{22}+23a^{21}-1118282a^{20}-230a^{19}+2043184a^{18}+1311a^{17}-2779568a^{16}-4692a^{15}+2782186a^{14}+10948a^{13}-2006731a^{12}-16745a^{11}+1010361a^{10}+16456a^{9}-338844a^{8}-9911a^{7}+70499a^{6}+3366a^{5}-8041a^{4}-561a^{3}+374a^{2}+34a-1$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476186a^{21}-9126996a^{19}+11561024a^{17}-a^{16}-10995872a^{15}+16a^{14}+7699384a^{13}-104a^{12}-3853955a^{11}+352a^{10}+1321617a^{9}-660a^{8}-291972a^{7}+672a^{6}+37932a^{5}-336a^{4}-2494a^{3}+64a^{2}+58a-3$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+5440a^{4}-256a^{2}+2$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{36}-36a^{34}-a^{33}+594a^{32}+33a^{31}-5951a^{30}-495a^{29}+40425a^{28}+4466a^{27}-196910a^{26}-27027a^{25}+709254a^{24}+115829a^{23}-1919971a^{22}-361767a^{21}+3930378a^{20}+834670a^{19}-6072230a^{18}-1426368a^{17}+7009916a^{16}+1793126a^{15}-5933018a^{14}-1630539a^{13}+3569657a^{12}+1041417a^{11}-1454375a^{10}-445510a^{9}+371020a^{8}+118197a^{7}-51676a^{6}-17089a^{5}+2930a^{4}+1080a^{3}-5a^{2}-20a-1$, $a^{37}-37a^{35}+629a^{33}-a^{32}-6511a^{31}+32a^{30}+45849a^{29}-464a^{28}-232406a^{27}+4031a^{26}+875133a^{25}-23374a^{24}-2490346a^{23}+95381a^{22}+5395479a^{21}-281358a^{20}-8896997a^{19}+606651a^{18}+11081654a^{17}-955435a^{16}-10265246a^{15}+1086588a^{14}+6893822a^{13}-871728a^{12}-3225872a^{11}+474617a^{10}+988338a^{9}-165022a^{8}-178828a^{7}+33320a^{6}+15729a^{5}-3360a^{4}-385a^{3}+126a^{2}-6a-1$, $a^{2}-2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a$, $a^{18}-18a^{16}+135a^{14}-545a^{12}+1275a^{10}-1728a^{8}+1275a^{6}-441a^{4}+54a^{2}-1$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-a^{15}-1136960a^{14}+15a^{13}+940576a^{12}-90a^{11}-537472a^{10}+275a^{9}+201552a^{8}-450a^{7}-45696a^{6}+378a^{5}+5440a^{4}-140a^{3}-256a^{2}+15a+3$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314610a^{9}-286824a^{7}+35854a^{5}-2114a^{3}+42a-1$, $a^{21}-21a^{19}+189a^{17}-a^{16}-952a^{15}+16a^{14}+2940a^{13}-104a^{12}-5733a^{11}+352a^{10}+7007a^{9}-660a^{8}-5148a^{7}+672a^{6}+2079a^{5}-336a^{4}-385a^{3}+64a^{2}+21a-2$, $a^{38}-37a^{36}+629a^{34}-a^{33}-6512a^{32}+32a^{31}+45880a^{30}-464a^{29}-232840a^{28}+4032a^{27}+878760a^{26}-23400a^{25}-2510496a^{24}+95679a^{23}+5473908a^{22}-283338a^{21}-9116580a^{20}+615086a^{19}+11528516a^{18}-979487a^{17}-10925151a^{16}+1133068a^{15}+7591777a^{14}-931903a^{13}-3740726a^{12}+524731a^{11}+1241405a^{10}-189387a^{9}-255486a^{8}+38434a^{7}+28105a^{6}-2926a^{5}-1155a^{4}-175a^{3}+20a$, $a^{37}-37a^{35}+629a^{33}-a^{32}-6512a^{31}+32a^{30}+45880a^{29}-464a^{28}-232841a^{27}+4032a^{26}+878787a^{25}-23400a^{24}-2510820a^{23}+95680a^{22}+5476185a^{21}-283360a^{20}-9126975a^{19}+615296a^{18}+11560835a^{17}-980628a^{16}-10994920a^{15}+1136960a^{14}+7696444a^{13}-940576a^{12}-3848222a^{11}+537472a^{10}+1314610a^{9}-201552a^{8}-286824a^{7}+45696a^{6}+35854a^{5}-5440a^{4}-2114a^{3}+256a^{2}+42a-3$, $a^{36}-a^{35}-35a^{34}+35a^{33}+560a^{32}-560a^{31}-5425a^{30}+5425a^{29}+35526a^{28}-35526a^{27}-166284a^{26}+166283a^{25}+573624a^{24}-573599a^{23}-1482327a^{22}+1482052a^{21}+2888270a^{20}-2886521a^{19}-4238443a^{18}+4231337a^{17}+4646015a^{16}-4626787a^{15}-3744743a^{14}+3709708a^{13}+2164474a^{12}-2122003a^{11}-863344a^{10}+830311a^{9}+223557a^{8}-208190a^{7}-33762a^{6}+30011a^{5}+2370a^{4}-2005a^{3}-36a^{2}+29a+1$, $a^{38}-37a^{36}+629a^{34}-a^{33}-6512a^{32}+32a^{31}+45880a^{30}-464a^{29}-232840a^{28}+4032a^{27}+878760a^{26}-23400a^{25}-2510496a^{24}+95680a^{23}+5473908a^{22}-283361a^{21}-9116580a^{20}+615317a^{19}+11528516a^{18}-980817a^{17}-10925151a^{16}+1137912a^{15}+7591777a^{14}-943516a^{13}-3740726a^{12}+543204a^{11}+1241405a^{10}-208549a^{9}-255486a^{8}+50809a^{7}+28105a^{6}-7469a^{5}-1155a^{4}+616a^{3}-22a$, $a^{34}-34a^{32}+527a^{30}-4930a^{28}-a^{27}+31059a^{26}+27a^{25}-139230a^{24}-324a^{23}+457470a^{22}+2277a^{21}-1118259a^{20}-10395a^{19}+2042955a^{18}+32319a^{17}-2778275a^{16}-69768a^{15}+2777630a^{14}+104652a^{13}-1996345a^{12}-107406a^{11}+995006a^{10}+72929a^{9}-324510a^{8}-30879a^{7}+62580a^{6}+7344a^{5}-5775a^{4}-789a^{3}+126a^{2}+18a$, $a^{36}-36a^{34}+594a^{32}-5952a^{30}+40455a^{28}-197316a^{26}+712530a^{24}-1937520a^{22}+3996135a^{20}-6249100a^{18}+7354710a^{16}-6418656a^{14}+4056234a^{12}-1790712a^{10}+523260a^{8}-93024a^{6}+8721a^{4}-324a^{2}+3$, $a^{7}-7a^{5}+14a^{3}-7a-1$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-1$, $a^{37}-37a^{35}+629a^{33}-a^{32}-6512a^{31}+31a^{30}+45880a^{29}-434a^{28}-232840a^{27}+3627a^{26}+878761a^{25}-20150a^{24}-2510521a^{23}+78429a^{22}+5474183a^{21}-219583a^{20}-9118330a^{19}+446862a^{18}+11535642a^{17}-659906a^{16}-10944548a^{15}+697970a^{14}+7627596a^{13}-514944a^{12}-3785368a^{11}+253342a^{10}+1278090a^{9}-77108a^{8}-274482a^{7}+12754a^{6}+33817a^{5}-910a^{4}-1995a^{3}+21a^{2}+35a-1$, $a^{38}-a^{37}-37a^{36}+36a^{35}+629a^{34}-595a^{33}-6511a^{32}+5984a^{31}+45849a^{30}-40919a^{29}-232406a^{28}+201347a^{27}+875133a^{26}-735904a^{25}-2490346a^{24}+2032900a^{23}+5395479a^{22}-4277471a^{21}-8896997a^{20}+6855541a^{19}+11081654a^{18}-8309004a^{17}-10265245a^{16}+7501352a^{15}+6893807a^{14}-4919289a^{13}-3225782a^{12}+2252589a^{11}+988063a^{10}-676127a^{9}-178378a^{8}+119116a^{7}+15351a^{6}-9611a^{5}-245a^{4}+35a^{3}-20a^{2}+20a-1$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{36}-36a^{34}+594a^{32}-a^{31}-5952a^{30}+31a^{29}+40455a^{28}-434a^{27}-197315a^{26}+3627a^{25}+712504a^{24}-20150a^{23}-1937221a^{22}+78429a^{21}+3994133a^{20}-219583a^{19}-6240455a^{18}+446862a^{17}+7329517a^{16}-659906a^{15}-6368284a^{14}+697970a^{13}+3987386a^{12}-514944a^{11}-1727858a^{10}+253342a^{9}+486740a^{8}-77108a^{7}-80682a^{6}+12754a^{5}+6685a^{4}-910a^{3}-210a^{2}+20a$, $a^{37}-36a^{35}+594a^{33}-a^{32}-5952a^{31}+31a^{30}+40455a^{29}-434a^{28}-197315a^{27}+3627a^{26}+712504a^{25}-20150a^{24}-1937221a^{23}+78429a^{22}+3994133a^{21}-219583a^{20}-6240455a^{19}+446862a^{18}+7329517a^{17}-659906a^{16}-6368284a^{15}+697970a^{14}+3987386a^{13}-514944a^{12}-1727858a^{11}+253342a^{10}+486740a^{9}-77108a^{8}-80682a^{7}+12754a^{6}+6685a^{5}-910a^{4}-210a^{3}+20a^{2}+1$, $a^{9}-9a^{7}-a^{6}+27a^{5}+6a^{4}-29a^{3}-9a^{2}+6a+1$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a$, $a^{35}-a^{34}-34a^{33}+33a^{32}+528a^{31}-495a^{30}-4961a^{29}+4466a^{28}+31493a^{27}-27027a^{26}-142857a^{25}+115830a^{24}+477620a^{23}-361791a^{22}-1196689a^{21}+834921a^{20}+2262559a^{19}-1427869a^{18}-3225327a^{17}+1798788a^{16}+3438504a^{15}-1644561a^{14}-2697359a^{13}+1064427a^{12}+1516034a^{11}-470184a^{10}-585507a^{9}+134837a^{8}+145456a^{7}-23654a^{6}-20846a^{5}+2406a^{4}+1408a^{3}-124a^{2}-24a$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476186a^{21}-9126996a^{19}+11561024a^{17}-10995872a^{15}+7699384a^{13}-3853955a^{11}+1321617a^{9}-291972a^{7}+37932a^{5}-2494a^{3}+58a-1$ (assuming GRH)
Regulator:		\( 536916737153376600000000 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{39}\cdot(2\pi)^{0}\cdot 536916737153376600000000 \cdot 1}{2\cdot\sqrt{1287743804278744050410620426954739687963064854495168753870500853746064161}}\cr\approx \mathstrut & 0.130056494208649 \end{aligned}\] (assuming GRH)

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

3.3.6241.1, 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	$39$	$39$	$39$	$39$	$39$	$39$	${\href{/padicField/17.13.0.1}{13} }^{3}$	$39$	${\href{/padicField/23.3.0.1}{3} }^{13}$	$39$	$39$	$39$	${\href{/padicField/41.13.0.1}{13} }^{3}$	$39$	$39$	$39$	$39$

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
\(79\)		Deg $39$	$39$	$1$	$38$