Number field 46.0.1846885917150184816801956108608187290274879518836200811774628627713584413482455417169884083661381807955707.1

• Label: 46.0.184...707.1
• Degree: $46$
• Signature: $[0, 23]$
• Discriminant: $-1.847\times 10^{105}$
• Root discriminant: \(194.27\)
• Ramified primes: $3,139$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{46}$ (as 46T1)

Normalized defining polynomial

x^46 - x^45 + 67*x^44 - 228*x^43 + 2873*x^42 - 12090*x^41 + 90627*x^40 - 396783*x^39 + 2197181*x^38 - 9084879*x^37 + 41123762*x^36 - 155774062*x^35 + 601546586*x^34 - 2049772191*x^33 + 6885710712*x^32 - 21001194790*x^31 + 62098669354*x^30 - 169080636457*x^29 + 441940766812*x^28 - 1071462326145*x^27 + 2478764771335*x^26 - 5333328598630*x^25 + 10888318672634*x^24 - 20655166588685*x^23 + 36991730149635*x^22 - 61402831326766*x^21 + 95739989836944*x^20 - 137578715879678*x^19 + 184445470774510*x^18 - 226065763101687*x^17 + 257516397067309*x^16 - 267113581606853*x^15 + 259035364360001*x^14 - 228631741303908*x^13 + 190505773653737*x^12 - 142852907594579*x^11 + 102011442664694*x^10 - 64133248169993*x^9 + 39534514646718*x^8 - 20666502848870*x^7 + 11038378291251*x^6 - 4490765253972*x^5 + 2041881142688*x^4 - 598697077332*x^3 + 238557870323*x^2 - 33007559079*x + 4088451481

Invariants

Degree:		$46$
Signature:		$[0, 23]$
Discriminant:		-1846885917150184816801956108608187290274879518836200811774628627713584413482455417169884083661381807955707
Root discriminant:		\(194.27\)
Galois root discriminant:		$3^{1/2}139^{22/23}\approx 194.26782195319768$
Ramified primes:		\(3\), \(139\)
Discriminant root field:		\(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:		$46$
This field is Galois and abelian over $\Q$.
Conductor:		\(417=3\cdot 139\)
Dirichlet character group:		$\lbrace$$\chi_{417}(1,·)$, $\chi_{417}(131,·)$, $\chi_{417}(394,·)$, $\chi_{417}(140,·)$, $\chi_{417}(194,·)$, $\chi_{417}(145,·)$, $\chi_{417}(403,·)$, $\chi_{417}(407,·)$, $\chi_{417}(409,·)$, $\chi_{417}(284,·)$, $\chi_{417}(34,·)$, $\chi_{417}(91,·)$, $\chi_{417}(44,·)$, $\chi_{417}(173,·)$, $\chi_{417}(175,·)$, $\chi_{417}(52,·)$, $\chi_{417}(55,·)$, $\chi_{417}(184,·)$, $\chi_{417}(314,·)$, $\chi_{417}(191,·)$, $\chi_{417}(64,·)$, $\chi_{417}(65,·)$, $\chi_{417}(322,·)$, $\chi_{417}(323,·)$, $\chi_{417}(196,·)$, $\chi_{417}(268,·)$, $\chi_{417}(202,·)$, $\chi_{417}(203,·)$, $\chi_{417}(77,·)$, $\chi_{417}(79,·)$, $\chi_{417}(80,·)$, $\chi_{417}(341,·)$, $\chi_{417}(343,·)$, $\chi_{417}(218,·)$, $\chi_{417}(335,·)$, $\chi_{417}(355,·)$, $\chi_{417}(100,·)$, $\chi_{417}(230,·)$, $\chi_{417}(358,·)$, $\chi_{417}(106,·)$, $\chi_{417}(239,·)$, $\chi_{417}(112,·)$, $\chi_{417}(116,·)$, $\chi_{417}(245,·)$, $\chi_{417}(251,·)$, $\chi_{417}(125,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4194304}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{43}a^{31}+\frac{20}{43}a^{30}-\frac{11}{43}a^{29}-\frac{20}{43}a^{28}+\frac{11}{43}a^{27}-\frac{8}{43}a^{26}+\frac{12}{43}a^{24}-\frac{19}{43}a^{23}-\frac{16}{43}a^{22}-\frac{19}{43}a^{21}+\frac{9}{43}a^{20}+\frac{1}{43}a^{19}-\frac{9}{43}a^{18}-\frac{19}{43}a^{17}+\frac{14}{43}a^{16}-\frac{12}{43}a^{15}+\frac{7}{43}a^{14}+\frac{4}{43}a^{13}+\frac{10}{43}a^{12}-\frac{17}{43}a^{11}+\frac{3}{43}a^{10}-\frac{4}{43}a^{9}-\frac{18}{43}a^{8}-\frac{6}{43}a^{7}+\frac{21}{43}a^{6}+\frac{18}{43}a^{5}+\frac{17}{43}a^{4}-\frac{16}{43}a^{3}+\frac{13}{43}a^{2}-\frac{10}{43}a$, $\frac{1}{43}a^{32}+\frac{19}{43}a^{30}-\frac{15}{43}a^{29}-\frac{19}{43}a^{28}-\frac{13}{43}a^{27}-\frac{12}{43}a^{26}+\frac{12}{43}a^{25}-\frac{1}{43}a^{24}+\frac{20}{43}a^{23}+\frac{2}{43}a^{21}-\frac{7}{43}a^{20}+\frac{14}{43}a^{19}-\frac{11}{43}a^{18}+\frac{7}{43}a^{17}+\frac{9}{43}a^{16}-\frac{11}{43}a^{15}-\frac{7}{43}a^{14}+\frac{16}{43}a^{13}-\frac{2}{43}a^{12}-\frac{1}{43}a^{11}-\frac{21}{43}a^{10}+\frac{19}{43}a^{9}+\frac{10}{43}a^{8}+\frac{12}{43}a^{7}-\frac{15}{43}a^{6}+\frac{1}{43}a^{5}-\frac{12}{43}a^{4}-\frac{11}{43}a^{3}-\frac{12}{43}a^{2}-\frac{15}{43}a$, $\frac{1}{43}a^{33}-\frac{8}{43}a^{30}+\frac{18}{43}a^{29}-\frac{20}{43}a^{28}-\frac{6}{43}a^{27}-\frac{8}{43}a^{26}-\frac{1}{43}a^{25}+\frac{7}{43}a^{24}+\frac{17}{43}a^{23}+\frac{5}{43}a^{22}+\frac{10}{43}a^{21}+\frac{15}{43}a^{20}+\frac{13}{43}a^{19}+\frac{6}{43}a^{18}-\frac{17}{43}a^{17}-\frac{19}{43}a^{16}+\frac{6}{43}a^{15}+\frac{12}{43}a^{14}+\frac{8}{43}a^{13}-\frac{19}{43}a^{12}+\frac{1}{43}a^{11}+\frac{5}{43}a^{10}+\frac{10}{43}a^{8}+\frac{13}{43}a^{7}-\frac{11}{43}a^{6}-\frac{10}{43}a^{5}+\frac{10}{43}a^{4}-\frac{9}{43}a^{3}-\frac{4}{43}a^{2}+\frac{18}{43}a$, $\frac{1}{43}a^{34}+\frac{6}{43}a^{30}+\frac{21}{43}a^{29}+\frac{6}{43}a^{28}-\frac{6}{43}a^{27}+\frac{21}{43}a^{26}+\frac{7}{43}a^{25}-\frac{16}{43}a^{24}-\frac{18}{43}a^{23}+\frac{11}{43}a^{22}-\frac{8}{43}a^{21}-\frac{1}{43}a^{20}+\frac{14}{43}a^{19}-\frac{3}{43}a^{18}+\frac{1}{43}a^{17}-\frac{11}{43}a^{16}+\frac{2}{43}a^{15}+\frac{21}{43}a^{14}+\frac{13}{43}a^{13}-\frac{5}{43}a^{12}-\frac{2}{43}a^{11}-\frac{19}{43}a^{10}+\frac{21}{43}a^{9}-\frac{2}{43}a^{8}-\frac{16}{43}a^{7}-\frac{14}{43}a^{6}-\frac{18}{43}a^{5}-\frac{2}{43}a^{4}-\frac{3}{43}a^{3}-\frac{7}{43}a^{2}+\frac{6}{43}a$, $\frac{1}{43}a^{35}-\frac{13}{43}a^{30}-\frac{14}{43}a^{29}-\frac{15}{43}a^{28}-\frac{2}{43}a^{27}+\frac{12}{43}a^{26}-\frac{16}{43}a^{25}-\frac{4}{43}a^{24}-\frac{4}{43}a^{23}+\frac{2}{43}a^{22}-\frac{16}{43}a^{21}+\frac{3}{43}a^{20}-\frac{9}{43}a^{19}+\frac{12}{43}a^{18}+\frac{17}{43}a^{17}+\frac{4}{43}a^{16}+\frac{7}{43}a^{15}+\frac{14}{43}a^{14}+\frac{14}{43}a^{13}-\frac{19}{43}a^{12}-\frac{3}{43}a^{11}+\frac{3}{43}a^{10}-\frac{21}{43}a^{9}+\frac{6}{43}a^{8}-\frac{21}{43}a^{7}-\frac{15}{43}a^{6}+\frac{19}{43}a^{5}-\frac{19}{43}a^{4}+\frac{3}{43}a^{3}+\frac{14}{43}a^{2}+\frac{17}{43}a$, $\frac{1}{4171}a^{36}-\frac{48}{4171}a^{35}-\frac{22}{4171}a^{34}+\frac{40}{4171}a^{33}+\frac{37}{4171}a^{32}-\frac{9}{4171}a^{31}+\frac{124}{4171}a^{30}+\frac{918}{4171}a^{29}-\frac{1212}{4171}a^{28}-\frac{1469}{4171}a^{27}+\frac{1762}{4171}a^{26}-\frac{2039}{4171}a^{25}+\frac{272}{4171}a^{24}-\frac{302}{4171}a^{23}-\frac{1852}{4171}a^{22}-\frac{1020}{4171}a^{21}+\frac{977}{4171}a^{20}-\frac{1832}{4171}a^{19}+\frac{11}{43}a^{18}+\frac{8}{43}a^{17}+\frac{1105}{4171}a^{16}-\frac{2043}{4171}a^{15}+\frac{763}{4171}a^{14}-\frac{1468}{4171}a^{13}+\frac{182}{4171}a^{12}+\frac{1846}{4171}a^{11}+\frac{1279}{4171}a^{10}+\frac{207}{4171}a^{9}-\frac{1287}{4171}a^{8}+\frac{1124}{4171}a^{7}-\frac{1971}{4171}a^{6}-\frac{1772}{4171}a^{5}+\frac{252}{4171}a^{4}+\frac{696}{4171}a^{3}-\frac{1053}{4171}a^{2}+\frac{381}{4171}a+\frac{33}{97}$, $\frac{1}{4171}a^{37}+\frac{2}{4171}a^{35}-\frac{46}{4171}a^{34}+\frac{17}{4171}a^{33}+\frac{21}{4171}a^{32}-\frac{17}{4171}a^{31}-\frac{211}{4171}a^{30}+\frac{2015}{4171}a^{29}-\frac{1736}{4171}a^{28}+\frac{23}{4171}a^{27}-\frac{1853}{4171}a^{26}-\frac{1085}{4171}a^{25}+\frac{435}{4171}a^{24}+\frac{239}{4171}a^{23}-\frac{1355}{4171}a^{22}-\frac{453}{4171}a^{21}-\frac{720}{4171}a^{20}-\frac{1800}{4171}a^{19}-\frac{15}{43}a^{18}-\frac{1805}{4171}a^{17}-\frac{219}{4171}a^{16}+\frac{87}{4171}a^{15}-\frac{152}{4171}a^{14}-\frac{636}{4171}a^{13}+\frac{591}{4171}a^{12}+\frac{744}{4171}a^{11}-\frac{1257}{4171}a^{10}+\frac{16}{4171}a^{9}+\frac{1040}{4171}a^{8}-\frac{1951}{4171}a^{7}+\frac{523}{4171}a^{6}-\frac{1772}{4171}a^{5}-\frac{1855}{4171}a^{4}+\frac{1703}{4171}a^{3}-\frac{208}{4171}a^{2}-\frac{760}{4171}a+\frac{32}{97}$, $\frac{1}{4171}a^{38}-\frac{47}{4171}a^{35}-\frac{36}{4171}a^{34}+\frac{38}{4171}a^{33}+\frac{6}{4171}a^{32}+\frac{1}{4171}a^{31}-\frac{949}{4171}a^{30}-\frac{1923}{4171}a^{29}-\frac{4}{97}a^{28}-\frac{2019}{4171}a^{27}+\frac{1211}{4171}a^{26}-\frac{1889}{4171}a^{25}+\frac{374}{4171}a^{24}+\frac{1286}{4171}a^{23}-\frac{629}{4171}a^{22}+\frac{1126}{4171}a^{21}-\frac{1426}{4171}a^{20}+\frac{366}{4171}a^{19}+\frac{1299}{4171}a^{18}+\frac{169}{4171}a^{17}+\frac{302}{4171}a^{16}+\frac{248}{4171}a^{15}+\frac{457}{4171}a^{14}-\frac{159}{4171}a^{13}-\frac{1560}{4171}a^{12}+\frac{580}{4171}a^{11}-\frac{1960}{4171}a^{10}+\frac{1693}{4171}a^{9}-\frac{1317}{4171}a^{8}-\frac{1046}{4171}a^{7}-\frac{1807}{4171}a^{6}+\frac{40}{4171}a^{5}-\frac{2002}{4171}a^{4}+\frac{1698}{4171}a^{3}+\frac{1637}{4171}a^{2}+\frac{905}{4171}a+\frac{31}{97}$, $\frac{1}{4171}a^{39}+\frac{36}{4171}a^{35}-\frac{26}{4171}a^{34}+\frac{1}{97}a^{33}-\frac{6}{4171}a^{32}-\frac{14}{4171}a^{31}+\frac{704}{4171}a^{30}+\frac{488}{4171}a^{29}+\frac{672}{4171}a^{28}-\frac{417}{4171}a^{27}-\frac{264}{4171}a^{26}+\frac{959}{4171}a^{25}-\frac{1450}{4171}a^{24}-\frac{176}{4171}a^{23}-\frac{1431}{4171}a^{22}-\frac{284}{4171}a^{21}-\frac{954}{4171}a^{20}-\frac{1579}{4171}a^{19}+\frac{363}{4171}a^{18}-\frac{280}{4171}a^{17}+\frac{1549}{4171}a^{16}-\frac{2056}{4171}a^{15}+\frac{685}{4171}a^{14}-\frac{37}{4171}a^{13}-\frac{372}{4171}a^{12}-\frac{1528}{4171}a^{11}-\frac{1535}{4171}a^{10}-\frac{318}{4171}a^{9}-\frac{1395}{4171}a^{8}+\frac{290}{4171}a^{7}+\frac{620}{4171}a^{6}-\frac{702}{4171}a^{5}+\frac{1320}{4171}a^{4}-\frac{1735}{4171}a^{3}-\frac{1832}{4171}a^{2}-\frac{1809}{4171}a-\frac{1}{97}$, $\frac{1}{4171}a^{40}-\frac{44}{4171}a^{35}-\frac{38}{4171}a^{34}+\frac{9}{4171}a^{33}+\frac{12}{4171}a^{32}-\frac{39}{4171}a^{31}-\frac{2036}{4171}a^{30}-\frac{366}{4171}a^{29}+\frac{1408}{4171}a^{28}-\frac{1797}{4171}a^{27}-\frac{199}{4171}a^{26}+\frac{174}{4171}a^{25}-\frac{1335}{4171}a^{24}+\frac{32}{4171}a^{23}-\frac{1609}{4171}a^{22}-\frac{221}{4171}a^{21}-\frac{861}{4171}a^{20}-\frac{1779}{4171}a^{19}+\frac{593}{4171}a^{18}-\frac{1846}{4171}a^{17}+\frac{1329}{4171}a^{16}+\frac{125}{4171}a^{15}+\frac{1110}{4171}a^{14}-\frac{98}{4171}a^{13}+\frac{941}{4171}a^{12}-\frac{1061}{4171}a^{11}-\frac{1063}{4171}a^{10}+\frac{2017}{4171}a^{9}+\frac{1808}{4171}a^{8}-\frac{1820}{4171}a^{7}-\frac{168}{4171}a^{6}-\frac{1430}{4171}a^{5}+\frac{57}{4171}a^{4}+\frac{1242}{4171}a^{3}-\frac{1537}{4171}a^{2}+\frac{1179}{4171}a-\frac{24}{97}$, $\frac{1}{754951}a^{41}-\frac{82}{754951}a^{40}+\frac{81}{754951}a^{39}-\frac{13}{754951}a^{38}-\frac{87}{754951}a^{37}-\frac{45}{754951}a^{36}+\frac{84}{754951}a^{35}-\frac{5062}{754951}a^{34}+\frac{1908}{754951}a^{33}-\frac{2287}{754951}a^{32}-\frac{3735}{754951}a^{31}+\frac{196659}{754951}a^{30}-\frac{8120}{17557}a^{29}+\frac{274310}{754951}a^{28}-\frac{359778}{754951}a^{27}+\frac{11259}{754951}a^{26}+\frac{136119}{754951}a^{25}-\frac{51897}{754951}a^{24}+\frac{175453}{754951}a^{23}+\frac{15680}{754951}a^{22}+\frac{11320}{754951}a^{21}-\frac{201402}{754951}a^{20}-\frac{175402}{754951}a^{19}-\frac{4201}{17557}a^{18}-\frac{72683}{754951}a^{17}+\frac{46934}{754951}a^{16}+\frac{3463}{754951}a^{15}-\frac{203228}{754951}a^{14}-\frac{111984}{754951}a^{13}-\frac{70319}{754951}a^{12}-\frac{202153}{754951}a^{11}-\frac{22648}{754951}a^{10}-\frac{205580}{754951}a^{9}+\frac{8334}{754951}a^{8}-\frac{67565}{754951}a^{7}+\frac{266209}{754951}a^{6}-\frac{225673}{754951}a^{5}-\frac{236839}{754951}a^{4}-\frac{230644}{754951}a^{3}-\frac{211685}{754951}a^{2}+\frac{248175}{754951}a+\frac{3905}{17557}$, $\frac{1}{3148900621}a^{42}-\frac{1080}{3148900621}a^{41}+\frac{365906}{3148900621}a^{40}-\frac{345111}{3148900621}a^{39}-\frac{174267}{3148900621}a^{38}+\frac{3521}{3148900621}a^{37}-\frac{160803}{3148900621}a^{36}-\frac{8242401}{3148900621}a^{35}-\frac{9159422}{3148900621}a^{34}+\frac{17701259}{3148900621}a^{33}+\frac{35675001}{3148900621}a^{32}-\frac{23353416}{3148900621}a^{31}-\frac{535620059}{3148900621}a^{30}-\frac{1140706132}{3148900621}a^{29}-\frac{589728867}{3148900621}a^{28}+\frac{1548094903}{3148900621}a^{27}-\frac{1208494822}{3148900621}a^{26}-\frac{239288031}{3148900621}a^{25}-\frac{101833557}{3148900621}a^{24}-\frac{1461808717}{3148900621}a^{23}-\frac{1348967666}{3148900621}a^{22}+\frac{497010059}{3148900621}a^{21}+\frac{837081148}{3148900621}a^{20}+\frac{1569989593}{3148900621}a^{19}-\frac{229059662}{3148900621}a^{18}+\frac{284413212}{3148900621}a^{17}-\frac{965191573}{3148900621}a^{16}+\frac{773062559}{3148900621}a^{15}-\frac{938655082}{3148900621}a^{14}+\frac{8323403}{32462893}a^{13}+\frac{1090676115}{3148900621}a^{12}+\frac{1307180107}{3148900621}a^{11}-\frac{1252064420}{3148900621}a^{10}-\frac{1544894088}{3148900621}a^{9}+\frac{78250572}{3148900621}a^{8}+\frac{430129755}{3148900621}a^{7}+\frac{949130111}{3148900621}a^{6}+\frac{15914384}{32462893}a^{5}-\frac{1036222741}{3148900621}a^{4}-\frac{45332869}{3148900621}a^{3}+\frac{201805490}{3148900621}a^{2}-\frac{25110844}{73230247}a+\frac{99565}{1703029}$, $\frac{1}{872245472017}a^{43}-\frac{28}{872245472017}a^{42}-\frac{157117}{872245472017}a^{41}-\frac{2180150}{20284778419}a^{40}-\frac{59148455}{872245472017}a^{39}-\frac{78808445}{872245472017}a^{38}+\frac{4556842}{872245472017}a^{37}-\frac{67597240}{872245472017}a^{36}+\frac{6346546270}{872245472017}a^{35}-\frac{1512197731}{872245472017}a^{34}-\frac{1422340932}{872245472017}a^{33}-\frac{5692603847}{872245472017}a^{32}+\frac{680127500}{872245472017}a^{31}+\frac{268592083170}{872245472017}a^{30}+\frac{376592597020}{872245472017}a^{29}+\frac{105641415806}{872245472017}a^{28}+\frac{341207226333}{872245472017}a^{27}+\frac{135184036193}{872245472017}a^{26}-\frac{215832486530}{872245472017}a^{25}-\frac{28905928772}{872245472017}a^{24}-\frac{1800107548}{872245472017}a^{23}-\frac{230602990317}{872245472017}a^{22}+\frac{29569885454}{872245472017}a^{21}+\frac{118124720176}{872245472017}a^{20}+\frac{258286980701}{872245472017}a^{19}-\frac{140086911490}{872245472017}a^{18}-\frac{213495337626}{872245472017}a^{17}-\frac{380749598044}{872245472017}a^{16}+\frac{71715087303}{872245472017}a^{15}-\frac{89096966057}{872245472017}a^{14}-\frac{371111498522}{872245472017}a^{13}-\frac{152587111225}{872245472017}a^{12}+\frac{283982040467}{872245472017}a^{11}+\frac{89945893939}{872245472017}a^{10}-\frac{355506776695}{872245472017}a^{9}+\frac{164253106947}{872245472017}a^{8}+\frac{156600988842}{872245472017}a^{7}-\frac{62522011693}{872245472017}a^{6}+\frac{422674303688}{872245472017}a^{5}+\frac{338313095185}{872245472017}a^{4}-\frac{331591991828}{872245472017}a^{3}-\frac{260266139061}{872245472017}a^{2}+\frac{2834202868}{20284778419}a+\frac{44793373}{471739033}$, $\frac{1}{16\!\cdots\!83}a^{44}+\frac{76\!\cdots\!71}{16\!\cdots\!83}a^{43}-\frac{59\!\cdots\!74}{16\!\cdots\!83}a^{42}+\frac{45\!\cdots\!82}{16\!\cdots\!83}a^{41}-\frac{28\!\cdots\!54}{16\!\cdots\!83}a^{40}+\frac{10\!\cdots\!80}{16\!\cdots\!83}a^{39}+\frac{19\!\cdots\!31}{16\!\cdots\!83}a^{38}-\frac{44\!\cdots\!63}{16\!\cdots\!83}a^{37}-\frac{27\!\cdots\!83}{38\!\cdots\!81}a^{36}-\frac{18\!\cdots\!61}{16\!\cdots\!83}a^{35}-\frac{38\!\cdots\!56}{16\!\cdots\!83}a^{34}-\frac{19\!\cdots\!06}{16\!\cdots\!83}a^{33}+\frac{86\!\cdots\!31}{16\!\cdots\!83}a^{32}+\frac{72\!\cdots\!66}{16\!\cdots\!83}a^{31}-\frac{75\!\cdots\!26}{16\!\cdots\!83}a^{30}-\frac{61\!\cdots\!14}{16\!\cdots\!83}a^{29}+\frac{38\!\cdots\!44}{16\!\cdots\!83}a^{28}-\frac{26\!\cdots\!38}{16\!\cdots\!83}a^{27}+\frac{20\!\cdots\!70}{16\!\cdots\!83}a^{26}+\frac{56\!\cdots\!88}{16\!\cdots\!83}a^{25}-\frac{72\!\cdots\!66}{16\!\cdots\!83}a^{24}+\frac{21\!\cdots\!67}{16\!\cdots\!83}a^{23}-\frac{47\!\cdots\!10}{16\!\cdots\!83}a^{22}-\frac{37\!\cdots\!82}{16\!\cdots\!83}a^{21}+\frac{77\!\cdots\!89}{16\!\cdots\!83}a^{20}+\frac{35\!\cdots\!79}{16\!\cdots\!83}a^{19}+\frac{10\!\cdots\!68}{16\!\cdots\!83}a^{18}+\frac{31\!\cdots\!41}{16\!\cdots\!83}a^{17}+\frac{25\!\cdots\!72}{16\!\cdots\!83}a^{16}+\frac{73\!\cdots\!66}{16\!\cdots\!83}a^{15}+\frac{72\!\cdots\!26}{16\!\cdots\!83}a^{14}+\frac{62\!\cdots\!82}{16\!\cdots\!83}a^{13}-\frac{35\!\cdots\!63}{16\!\cdots\!83}a^{12}+\frac{62\!\cdots\!59}{16\!\cdots\!83}a^{11}+\frac{33\!\cdots\!73}{16\!\cdots\!83}a^{10}+\frac{71\!\cdots\!35}{16\!\cdots\!83}a^{9}-\frac{39\!\cdots\!68}{16\!\cdots\!83}a^{8}+\frac{14\!\cdots\!36}{16\!\cdots\!83}a^{7}+\frac{30\!\cdots\!80}{16\!\cdots\!83}a^{6}+\frac{14\!\cdots\!68}{16\!\cdots\!83}a^{5}+\frac{48\!\cdots\!82}{16\!\cdots\!83}a^{4}-\frac{78\!\cdots\!25}{16\!\cdots\!83}a^{3}+\frac{32\!\cdots\!82}{16\!\cdots\!83}a^{2}-\frac{17\!\cdots\!69}{38\!\cdots\!81}a+\frac{41\!\cdots\!35}{89\!\cdots\!67}$, $\frac{1}{14\!\cdots\!23}a^{45}-\frac{91\!\cdots\!19}{14\!\cdots\!23}a^{44}+\frac{18\!\cdots\!83}{14\!\cdots\!23}a^{43}-\frac{21\!\cdots\!29}{14\!\cdots\!23}a^{42}-\frac{84\!\cdots\!71}{14\!\cdots\!23}a^{41}+\frac{43\!\cdots\!36}{14\!\cdots\!23}a^{40}+\frac{13\!\cdots\!57}{14\!\cdots\!23}a^{39}+\frac{42\!\cdots\!22}{14\!\cdots\!23}a^{38}-\frac{15\!\cdots\!62}{14\!\cdots\!23}a^{37}+\frac{10\!\cdots\!57}{14\!\cdots\!23}a^{36}-\frac{58\!\cdots\!49}{14\!\cdots\!23}a^{35}-\frac{42\!\cdots\!71}{14\!\cdots\!23}a^{34}-\frac{17\!\cdots\!52}{14\!\cdots\!23}a^{33}+\frac{94\!\cdots\!39}{14\!\cdots\!23}a^{32}+\frac{54\!\cdots\!67}{14\!\cdots\!23}a^{31}-\frac{57\!\cdots\!83}{14\!\cdots\!23}a^{30}-\frac{71\!\cdots\!70}{14\!\cdots\!23}a^{29}-\frac{45\!\cdots\!85}{14\!\cdots\!23}a^{28}+\frac{16\!\cdots\!70}{79\!\cdots\!83}a^{27}-\frac{53\!\cdots\!44}{14\!\cdots\!23}a^{26}+\frac{20\!\cdots\!67}{14\!\cdots\!23}a^{25}-\frac{43\!\cdots\!79}{14\!\cdots\!23}a^{24}+\frac{61\!\cdots\!94}{14\!\cdots\!23}a^{23}-\frac{57\!\cdots\!68}{14\!\cdots\!23}a^{22}-\frac{19\!\cdots\!20}{14\!\cdots\!23}a^{21}+\frac{26\!\cdots\!27}{14\!\cdots\!23}a^{20}+\frac{30\!\cdots\!26}{51\!\cdots\!99}a^{19}-\frac{52\!\cdots\!91}{14\!\cdots\!23}a^{18}-\frac{50\!\cdots\!39}{14\!\cdots\!23}a^{17}+\frac{15\!\cdots\!44}{33\!\cdots\!61}a^{16}-\frac{46\!\cdots\!82}{14\!\cdots\!23}a^{15}-\frac{54\!\cdots\!80}{14\!\cdots\!23}a^{14}+\frac{30\!\cdots\!34}{14\!\cdots\!23}a^{13}-\frac{44\!\cdots\!77}{14\!\cdots\!23}a^{12}+\frac{22\!\cdots\!71}{14\!\cdots\!23}a^{11}-\frac{65\!\cdots\!43}{14\!\cdots\!23}a^{10}-\frac{28\!\cdots\!95}{14\!\cdots\!23}a^{9}-\frac{66\!\cdots\!58}{14\!\cdots\!23}a^{8}+\frac{36\!\cdots\!53}{14\!\cdots\!23}a^{7}+\frac{86\!\cdots\!46}{14\!\cdots\!23}a^{6}+\frac{63\!\cdots\!09}{14\!\cdots\!23}a^{5}-\frac{62\!\cdots\!02}{14\!\cdots\!23}a^{4}-\frac{65\!\cdots\!87}{14\!\cdots\!23}a^{3}+\frac{30\!\cdots\!34}{14\!\cdots\!23}a^{2}+\frac{25\!\cdots\!03}{33\!\cdots\!61}a+\frac{16\!\cdots\!72}{52\!\cdots\!21}$

Monogenic:		No
Index:		Not computed
Inessential primes:		$43$

Class group and class number

not computed

Unit group

Rank:		$22$
Torsion generator:		\( \frac{2950692613991133858345940774412126030098087472340910166481067602009971631944477977730853178707654955937892209383780599654282684198659242323135270028134374733201752130664953734385504}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{45} - \frac{3400325923146815893789237306916721532322727854760657467869520637851660802811506985772339178999980667803088989503558360437272522042020847752321691326760039476554938719340914539975216}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{44} + \frac{198322492232596282499753276960317591528661521303139117198644573491342172074586164034534128275141442057166708619318418801895177278613755511783502874439410702755680907705379528209203050}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{43} - \frac{702861051943242832630875182243283659043516972010851775732866380106486222440063803789966032971933273555059495490171957145594209659514542437955123437596751527883689042680239398562181745}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{42} + \frac{8591414030560745378247170801281488579826582138407738171147018829547909965034260435165173588857082002833174783904483234584327272883387832938119463769153982380924334457818822789773490843}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{41} - \frac{36992555832014910023862837582351032981657225321428485401405192755117532045454555962005226599968057202808128605990090278081769389054795725808391104478773577401432483762234789931936557880}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{40} + \frac{273305790778041759839298412880717579773304551043420069253954223244076444853573212737839140694140907773506760579186980766907273819096069929026765587601416738578924509131826529145722842698}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{39} - \frac{1213083348539190294009399338185620313790975883623071987223655846526081557846225531679904506025900538197934475017934008514862648574955635454156250562289463805740275190546506127005495335852}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{38} + \frac{6675023235435806445505922334261546374457274520843357501693649502669658722414356608338718568836156071939257742863016710643539196804844871975420451858671155323443523993079885580659346810816}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{37} - \frac{27845930470892950941585290450063410537730917047654553194633158456998056303851143501836008189131759791083963883070106546005238432109076534266321457660512812492214549196252658771355549386893}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{36} + \frac{125731882395471087433433633673888021716848849964628302893465666327879905710674906081837477501405700524691425489223562895993136711766878392726023615612384327894506959740465693576055254109728}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{35} - \frac{479276033995321615419773899481285380737778769495487996979673823059373525206381601118227785556967519471211638381373568705360867170166987318591987284551141613388343429007227470084354138801311}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{34} + \frac{1850344762753209539626544167729306451454789951257123435770081765852685378068219494747017065813438100370273432409263993526208327494593531338130555218092666865239340030720811882830604679479066}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{33} - \frac{6337535101209676007126359345043592115380275295811596393482955579409434597357586452581208201078786583359731094403196626891162216182467303796418730635840342804911110588900905402106424115876935}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{32} + \frac{21312163246440748853133901145232294696566400219803080959048313075517905076202163754896198721196825563536402240444591265665844840889387309592479427913361583810061160605413852811941592458183883}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{31} - \frac{65297869950210259905939347544046129723195390287176098499168255010827365050949396325751662671929273296414995403721765230328968396387768649502749352538976008770816723679673325591133041635288531}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{30} + \frac{193452337557994746753902419983455881809705784232668421178851985647319505980408096777660730275086804116358757777976066192669077272820120197794379075612686831625392935458084715910454542033600215}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{29} - \frac{529057260111019847907972379421784055657610731554571026578018250299789223596576250577684576996708603369312768506264243343358663146155861966818792338817387972610597133997964318844379605158203235}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{28} + \frac{1386485539879114399511655537869110959384407747651342243926872875065202012444535191669742324039551298338540894388499155762825277322571983913277170339788032238389673319018985628012234626418343795}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{27} - \frac{3376712591932639617830100186574392797746658356077908523043310470831089957857160202310503919550162112874012956581724871221354522042091642561612974255291063194562538414884407540465785181512443469}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{26} + \frac{7837174164354341970014582229168588891868585670150440268305790983751590296484463360291912127531141379845591862682227462008315154006530072887031264033016006729971986698290049730645919383535618292}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{25} - \frac{16944757839809119584019715579450606933054639834365278764496133562248212832145877719149493946658611667224154099956779736352582006175472924185880041397762112829082524412388545589584756716215051003}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{24} + \frac{34730119031099288541763516627351753191156530363271110524282627917714775792169528761067081877849612583236498459557006557009051538429784582353763589845956971397241436052668270838370925551339530903}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{23} - \frac{66243501387982055937980635783482511824809069633837596965538538532839841178156448311266652082897335816814594821621123505209013083283726002162362998493977076998714148187889474556557310220579010224}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{22} + \frac{119192957724780967275230011679882021516045558471018469151403193464571091841798304761885488930001365649181250537315267792237002122831251598980564945839725806633178907919626134149887030681110605670}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{21} - \frac{199074819499698837656384514025226436308458425308578246227690089729982893502229150374490298023258523472765921400049953546946478131292666980379946856019367384651353549731747325983392776628316151630}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{20} + \frac{312102009500665146320595557373725471728486000021765696440111482857567020011573382396167938006106906319360813788516773739015889304465765085645028099400946376061729931963994737951063913467970507823}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{19} - \frac{451718964931260993653974357722912962718817657633724471143083545736593624527923007761895120114978521817816013453124630762151047994474780943034618198055518243404204321433307752537851685861705573829}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{18} + \frac{609507115849992217927856930283245292749118138362171194061604644371906274041500082842680142880494916568470070914354095079192746027678565903730534560765860728682856535608029435612120733988830389707}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{17} - \frac{17517960026939676462990827904212326938239659117460288315076775017030572657364132578550660350514142118153280742875766542266157689536288617962440052399704721238647973088275224261905532767466472303}{1771167708860926077559008025186872812188134921175345563716718258809021323898740727077867001711385183427962042458483336514831822105353402779299934908727569507901826491597757408481570942828539} a^{16} + \frac{863915689921054820227712383250836264275954510649621172965871197992951032374452840025164080688419541710727802291093418535819671607589725955092445145596040906940554959944076556896966447046200080458}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{15} - \frac{903671025974790407367987462740296203841181417578482519201393124056336032138370711523676202840067115656863802957480009542555360499328567661685375362801466667644319547396577051870747001418368216676}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{14} + \frac{880889309307104074273260232679831462141904651993780687994081788264362895638122672040401557641601268576702855829239758867374347937758149282152974165950740891936346207798552997264668144485590180116}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{13} - \frac{783077246704844370610236562849515405714505298285528784917263096173113503116565838333486001371750969431406445494297973811644909933638328774910379349407952017334658440483126355007347667672713820471}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{12} + \frac{654132440321656242928477396296635820784737598865661586998921043669168672005160883662773320533702096683370548513944986763925396854260078659498177224061204037051503918865836794262739028066140012126}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{11} - \frac{493847188499373285198560036656239312930812589504037834618745384986932402149990227938954020562123823199509840602960063522010400350417295698082073463792365047150807927309811896671869017606567832888}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{10} + \frac{352376014827827495560050948918412563508524529320241908091382789866817677689137620245377194689372892287333121137150956630312300413248188605636233910838043758640903094964353392127468606946034013293}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{9} - \frac{222661005889285766440998496241656454953619302071032574332706102098903891823655639868411394062823350279460739794653894818223017270474241647378727684572798769581486618174222741428207391474658374925}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{8} + \frac{135942286246373866453258971911581641209034778033595639699831716630691391845102888666087141632075561504388321771845309376816044227095394310438256965088238328674194906833400035950249914732171468371}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{7} - \frac{71194924557711519139690326213863439676205376769405583531577874926076404339537208036642815445400353774726284068925754096208389520657132233366429118377484604431527089137694642446729008564306637806}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{6} + \frac{37420309858801108041035237145117015407595256391760218809512778536045112958550090744020807347009820200736402033121692415654640030182343175042407601799944767084569914187139474742134780008335375593}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{5} - \frac{15202286653761294924356196784738237649616068488118338770385323974408474065240874270248652438252575849745296725913079446486207591812456271726272802488615399555575990761766720979281346195260177846}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{4} + \frac{6626157480918000428699203010669279745149907392435281490511554928293173547635300996014840982001469604200675232140581597064258797714199309460695943441186516161877300561071368195519219424240601320}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{3} - \frac{1741815007477255699361081639188043831814351208989996478177313108869009034815183550027368730189230851822787035123979461463749201403248456545987290230830405513891434518285951643086028764479705360}{76160211481019821335037345083035530924089801610539859239818885128787916927645851264348281073589562887402367825714783470137768350530196319509897201075285488839778539138703568564707550541627177} a^{2} + \frac{16488399439315878701185898472209227352429073313973233140746389529857617163587685620932660258402047674947718146545378798564072569895703692697700534457651303533160206213263879491311745294665159}{1771167708860926077559008025186872812188134921175345563716718258809021323898740727077867001711385183427962042458483336514831822105353402779299934908727569507901826491597757408481570942828539} a - \frac{7527347259499127859683678030758668477257866443649504403671332275194525283563611375166886291433347631360374466712023525736748386832076325910382569779450395870216787761231349114607210429}{27700031417414899322172127823882529397227677408475712980978061944746271154638506233525703409571091841353154352582589207469883519265469773373890538288853310206312483251712632090232729279} \)  (order $6$)
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

$C_{46}$ (as 46T1):

A cyclic group of order 46

The 46 conjugacy class representatives for $C_{46}$

Character table for $C_{46}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 23.23.140063703503689367173618364344202364099995564521.1

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	$46$	R	$46$	$23^{2}$	$46$	$23^{2}$	$46$	$23^{2}$	$46$	$46$	$23^{2}$	$23^{2}$	$46$	${\href{/padicField/43.1.0.1}{1} }^{46}$	$46$	$46$	$46$

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
\(3\)		Deg $46$	$2$	$23$	$23$
\(139\)	139.23.22.1	$x^{23} + 139$	$23$	$1$	$22$	$C_{23}$	$[\ ]_{23}$
	139.23.22.1	$x^{23} + 139$	$23$	$1$	$22$	$C_{23}$	$[\ ]_{23}$