Normalized defining polynomial
\( x^{40} - 2 x^{39} + 49 x^{38} - 66 x^{37} + 1410 x^{36} - 1580 x^{35} + 25227 x^{34} - 21838 x^{33} + \cdots + 7921 \)
Invariants
| Degree: | $40$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(809451525758004789010365777553377865197930569782630809600000000000000000000\)
\(\medspace = 2^{60}\cdot 3^{20}\cdot 5^{20}\cdot 11^{32}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(74.59\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2^{3/2}3^{1/2}5^{1/2}11^{4/5}\approx 74.59415027789584$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $40$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1320=2^{3}\cdot 3\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(389,·)$, $\chi_{1320}(641,·)$, $\chi_{1320}(521,·)$, $\chi_{1320}(269,·)$, $\chi_{1320}(401,·)$, $\chi_{1320}(529,·)$, $\chi_{1320}(661,·)$, $\chi_{1320}(89,·)$, $\chi_{1320}(1049,·)$, $\chi_{1320}(1181,·)$, $\chi_{1320}(581,·)$, $\chi_{1320}(289,·)$, $\chi_{1320}(1189,·)$, $\chi_{1320}(169,·)$, $\chi_{1320}(301,·)$, $\chi_{1320}(49,·)$, $\chi_{1320}(949,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(829,·)$, $\chi_{1320}(181,·)$, $\chi_{1320}(449,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(709,·)$, $\chi_{1320}(929,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(1109,·)$, $\chi_{1320}(1241,·)$, $\chi_{1320}(221,·)$, $\chi_{1320}(1061,·)$, $\chi_{1320}(229,·)$, $\chi_{1320}(1169,·)$, $\chi_{1320}(421,·)$, $\chi_{1320}(749,·)$, $\chi_{1320}(509,·)$, $\chi_{1320}(881,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(889,·)$, $\chi_{1320}(1021,·)$, $\chi_{1320}(1301,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{524288}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{2}a^{30}-\frac{1}{2}a^{18}-\frac{1}{2}$, $\frac{1}{2}a^{31}-\frac{1}{2}a^{19}-\frac{1}{2}a$, $\frac{1}{2}a^{32}-\frac{1}{2}a^{20}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{33}-\frac{1}{2}a^{21}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{34}-\frac{1}{2}a^{22}-\frac{1}{2}a^{4}$, $\frac{1}{4378}a^{35}-\frac{367}{2189}a^{34}+\frac{615}{4378}a^{33}-\frac{403}{2189}a^{32}+\frac{12}{199}a^{31}-\frac{177}{4378}a^{30}+\frac{819}{2189}a^{29}+\frac{635}{2189}a^{28}-\frac{335}{2189}a^{27}+\frac{503}{2189}a^{26}-\frac{45}{2189}a^{25}+\frac{1082}{2189}a^{24}-\frac{1687}{4378}a^{23}-\frac{610}{2189}a^{22}+\frac{707}{4378}a^{21}+\frac{145}{2189}a^{20}+\frac{657}{2189}a^{19}-\frac{1887}{4378}a^{18}-\frac{602}{2189}a^{17}-\frac{969}{2189}a^{16}+\frac{767}{2189}a^{15}+\frac{459}{2189}a^{14}+\frac{706}{2189}a^{13}-\frac{1094}{2189}a^{12}+\frac{281}{2189}a^{11}-\frac{82}{2189}a^{10}-\frac{55}{199}a^{9}+\frac{1015}{2189}a^{8}+\frac{255}{2189}a^{7}+\frac{582}{2189}a^{6}+\frac{1577}{4378}a^{5}+\frac{877}{2189}a^{4}-\frac{139}{4378}a^{3}+\frac{142}{2189}a^{2}+\frac{549}{2189}a+\frac{1187}{4378}$, $\frac{1}{1449118}a^{36}-\frac{131}{1449118}a^{35}-\frac{170551}{1449118}a^{34}-\frac{207857}{1449118}a^{33}-\frac{104970}{724559}a^{32}+\frac{53943}{1449118}a^{31}+\frac{38297}{724559}a^{30}+\frac{244946}{724559}a^{29}-\frac{285075}{724559}a^{28}+\frac{319480}{724559}a^{27}+\frac{276996}{724559}a^{26}+\frac{81208}{724559}a^{25}+\frac{103633}{1449118}a^{24}-\frac{156015}{1449118}a^{23}+\frac{530289}{1449118}a^{22}-\frac{629}{4378}a^{21}+\frac{261023}{724559}a^{20}-\frac{387227}{1449118}a^{19}-\frac{213820}{724559}a^{18}+\frac{209543}{724559}a^{17}-\frac{333994}{724559}a^{16}-\frac{18620}{724559}a^{15}+\frac{54205}{724559}a^{14}+\frac{115975}{724559}a^{13}-\frac{291649}{724559}a^{12}-\frac{314408}{724559}a^{11}-\frac{297408}{724559}a^{10}-\frac{306886}{724559}a^{9}+\frac{49727}{724559}a^{8}+\frac{51464}{724559}a^{7}+\frac{541483}{1449118}a^{6}-\frac{115547}{1449118}a^{5}+\frac{133765}{1449118}a^{4}+\frac{113477}{1449118}a^{3}-\frac{161182}{724559}a^{2}-\frac{711411}{1449118}a+\frac{225446}{724559}$, $\frac{1}{1449118}a^{37}-\frac{35}{1449118}a^{35}-\frac{177417}{1449118}a^{34}+\frac{310509}{1449118}a^{33}+\frac{125107}{724559}a^{32}+\frac{174253}{1449118}a^{31}-\frac{10627}{65869}a^{30}+\frac{2110}{65869}a^{29}+\frac{274383}{724559}a^{28}+\frac{269292}{724559}a^{27}+\frac{348595}{724559}a^{26}+\frac{12895}{131738}a^{25}-\frac{31422}{65869}a^{24}-\frac{323399}{1449118}a^{23}+\frac{421259}{1449118}a^{22}-\frac{574557}{1449118}a^{21}-\frac{1061}{65869}a^{20}-\frac{177729}{1449118}a^{19}-\frac{186209}{724559}a^{18}+\frac{357106}{724559}a^{17}-\frac{292998}{724559}a^{16}+\frac{274239}{724559}a^{15}-\frac{107538}{724559}a^{14}+\frac{315561}{724559}a^{13}+\frac{337318}{724559}a^{12}-\frac{340563}{724559}a^{11}-\frac{314923}{724559}a^{10}-\frac{90416}{724559}a^{9}-\frac{22192}{724559}a^{8}-\frac{392647}{1449118}a^{7}-\frac{274423}{724559}a^{6}-\frac{165155}{1449118}a^{5}-\frac{241611}{1449118}a^{4}-\frac{675595}{1449118}a^{3}-\frac{23146}{65869}a^{2}+\frac{294193}{1449118}a-\frac{7000}{724559}$, $\frac{1}{22\!\cdots\!46}a^{38}-\frac{39\!\cdots\!33}{22\!\cdots\!46}a^{37}+\frac{13\!\cdots\!21}{56\!\cdots\!27}a^{36}+\frac{12\!\cdots\!49}{22\!\cdots\!46}a^{35}+\frac{58\!\cdots\!53}{11\!\cdots\!73}a^{34}+\frac{43\!\cdots\!31}{22\!\cdots\!46}a^{33}+\frac{15\!\cdots\!95}{22\!\cdots\!46}a^{32}-\frac{18\!\cdots\!53}{22\!\cdots\!46}a^{31}+\frac{51\!\cdots\!49}{22\!\cdots\!46}a^{30}+\frac{62\!\cdots\!02}{11\!\cdots\!73}a^{29}-\frac{35\!\cdots\!44}{11\!\cdots\!73}a^{28}-\frac{51\!\cdots\!11}{11\!\cdots\!73}a^{27}-\frac{14\!\cdots\!69}{22\!\cdots\!46}a^{26}-\frac{63\!\cdots\!61}{22\!\cdots\!46}a^{25}+\frac{39\!\cdots\!94}{11\!\cdots\!73}a^{24}+\frac{77\!\cdots\!63}{22\!\cdots\!46}a^{23}-\frac{69\!\cdots\!32}{11\!\cdots\!73}a^{22}+\frac{35\!\cdots\!03}{22\!\cdots\!46}a^{21}-\frac{53\!\cdots\!73}{22\!\cdots\!46}a^{20}-\frac{19\!\cdots\!63}{22\!\cdots\!46}a^{19}+\frac{94\!\cdots\!11}{22\!\cdots\!46}a^{18}-\frac{38\!\cdots\!46}{11\!\cdots\!73}a^{17}-\frac{42\!\cdots\!70}{11\!\cdots\!73}a^{16}-\frac{94\!\cdots\!70}{11\!\cdots\!73}a^{15}-\frac{33\!\cdots\!88}{11\!\cdots\!73}a^{14}+\frac{37\!\cdots\!81}{11\!\cdots\!73}a^{13}-\frac{25\!\cdots\!02}{11\!\cdots\!73}a^{12}-\frac{31\!\cdots\!05}{11\!\cdots\!73}a^{11}+\frac{39\!\cdots\!42}{11\!\cdots\!73}a^{10}+\frac{22\!\cdots\!79}{11\!\cdots\!73}a^{9}+\frac{45\!\cdots\!89}{22\!\cdots\!46}a^{8}-\frac{10\!\cdots\!89}{22\!\cdots\!46}a^{7}-\frac{50\!\cdots\!96}{11\!\cdots\!73}a^{6}-\frac{32\!\cdots\!99}{22\!\cdots\!46}a^{5}+\frac{15\!\cdots\!84}{11\!\cdots\!73}a^{4}+\frac{10\!\cdots\!29}{22\!\cdots\!46}a^{3}-\frac{86\!\cdots\!15}{20\!\cdots\!86}a^{2}+\frac{88\!\cdots\!01}{22\!\cdots\!46}a-\frac{56\!\cdots\!75}{22\!\cdots\!46}$, $\frac{1}{70\!\cdots\!86}a^{39}-\frac{10\!\cdots\!97}{63\!\cdots\!26}a^{38}+\frac{10\!\cdots\!95}{70\!\cdots\!86}a^{37}+\frac{36\!\cdots\!18}{35\!\cdots\!43}a^{36}+\frac{35\!\cdots\!99}{70\!\cdots\!86}a^{35}+\frac{57\!\cdots\!60}{35\!\cdots\!43}a^{34}-\frac{62\!\cdots\!29}{35\!\cdots\!43}a^{33}+\frac{55\!\cdots\!92}{31\!\cdots\!13}a^{32}+\frac{81\!\cdots\!68}{35\!\cdots\!43}a^{31}+\frac{78\!\cdots\!05}{70\!\cdots\!86}a^{30}-\frac{71\!\cdots\!19}{35\!\cdots\!43}a^{29}+\frac{13\!\cdots\!42}{31\!\cdots\!13}a^{28}-\frac{18\!\cdots\!01}{70\!\cdots\!86}a^{27}+\frac{26\!\cdots\!79}{70\!\cdots\!86}a^{26}-\frac{34\!\cdots\!51}{70\!\cdots\!86}a^{25}+\frac{11\!\cdots\!79}{35\!\cdots\!43}a^{24}+\frac{18\!\cdots\!97}{70\!\cdots\!86}a^{23}-\frac{13\!\cdots\!73}{35\!\cdots\!43}a^{22}-\frac{58\!\cdots\!32}{35\!\cdots\!43}a^{21}-\frac{95\!\cdots\!19}{35\!\cdots\!43}a^{20}-\frac{86\!\cdots\!98}{35\!\cdots\!43}a^{19}-\frac{29\!\cdots\!47}{70\!\cdots\!86}a^{18}-\frac{13\!\cdots\!29}{35\!\cdots\!43}a^{17}+\frac{12\!\cdots\!82}{35\!\cdots\!43}a^{16}+\frac{14\!\cdots\!59}{35\!\cdots\!43}a^{15}-\frac{11\!\cdots\!74}{35\!\cdots\!43}a^{14}+\frac{12\!\cdots\!45}{35\!\cdots\!43}a^{13}-\frac{48\!\cdots\!58}{35\!\cdots\!43}a^{12}+\frac{99\!\cdots\!06}{35\!\cdots\!43}a^{11}-\frac{66\!\cdots\!98}{35\!\cdots\!43}a^{10}-\frac{64\!\cdots\!47}{70\!\cdots\!86}a^{9}-\frac{26\!\cdots\!65}{70\!\cdots\!86}a^{8}+\frac{10\!\cdots\!31}{70\!\cdots\!86}a^{7}-\frac{14\!\cdots\!13}{35\!\cdots\!43}a^{6}+\frac{28\!\cdots\!01}{70\!\cdots\!86}a^{5}-\frac{13\!\cdots\!36}{35\!\cdots\!43}a^{4}-\frac{14\!\cdots\!01}{35\!\cdots\!43}a^{3}-\frac{67\!\cdots\!20}{35\!\cdots\!43}a^{2}-\frac{32\!\cdots\!01}{35\!\cdots\!43}a+\frac{32\!\cdots\!39}{78\!\cdots\!74}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
not computed
Unit group
| Rank: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{3918933250467393815726373552340254751386066348285554081591854231820052308338736758072053491263710247688352195958197181}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{39} - \frac{7841547983358750245014655336850950756147555162279952785952292441550075649904653736411476214574947695974198327522112947}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{38} + \frac{192100197403200606955179745353419028175593337647461021103141166833007755122026069817685038604920481063714436758257059527}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{37} - \frac{258946225984219720226167183378421728920483438423597021352315920269244103066626985863778631336461357603451072888074331319}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{36} + \frac{5529093253403011832863597038534926863863390213761005695826603474951205708219600234627153079597523065122194450318225343630}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{35} - \frac{6200701851281705523811996044410059739316307091101433395962270928199752455365102478715967480679354176724858329058538151672}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{34} + \frac{98959240650559277717794423671907567119010438606455251796863930619664890366288185949914374658851006668405168755887485491119}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{33} - \frac{171514357262610423485617456565152461580170286308397740239355693176396886611607438867511960279556586852630292059068619773457}{1056296022618440612519706108339350253476536521790931060633615299272877457813615843997750011747830739220621009845890748790262} a^{32} + \frac{1258041014087487056106698785259392163289116931058575262162973011205230618210457862815410711039506846687390797338402424285954}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{31} - \frac{930185479651915739206309097260549590787034042817943073667605152129562644273505174285850270352926818677486083432000366334780}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{30} + \frac{11610536633064092881799204679858778046485105234890691114907738250822939116059772519480097668119612703702999848296460053391388}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{29} - \frac{7413181476300748706209843919033726892057059280455546999720300631029007882590591206020075198268964282323460647147507444244040}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{28} + \frac{81515413310503357500642003925744693457400287398860564495344648187683905501363977543395085713692268425505319906462167927677057}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{27} - \frac{46918617314996480021934009156061412102902386526659109022611680961691191623301422080013616460523147754367542935770953112910150}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{26} + \frac{437440756448662554679977012705718873542425081860536474157225166571991141615749171936741542892254526704773244282434161716468839}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{25} - \frac{222558973279868661705206627341169560838194598844130448165375625848650970892807071792096013679305711671245931068899004686425197}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{24} + \frac{1825882123330881165909851988482951365321399836785555350588521250718202990523987450096153294149747420730356699015560726723654166}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{23} - \frac{821277920602938018881784652323301200175990503665769779386854594066332050167987553987025349642523265794471485908698258326503322}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{22} + \frac{535514076404951284797043301030891431267053306564303393760267009553676912400380117204500214042516893956221661542441447421063961}{48013455573565482387259368560879556976206205535951411846982513603312611718800720181715909624901397237300954992995034035921} a^{21} - \frac{4451420388938050593587766771055016444436710452501952653601267438959698423819263526739575972498420052912709139777055802136250621}{1056296022618440612519706108339350253476536521790931060633615299272877457813615843997750011747830739220621009845890748790262} a^{20} + \frac{14738074171922151825185789150820268377190607430031135432744041937111398260974192894541814989705688837439342160044998656313760302}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{19} - \frac{4718469559217229558880461019547084134195288781286427351851673839606473949295086904171681256045245745349923100908268923916294818}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{18} + \frac{28301691506080244443113125855753977786517755660653625704533537674782520830024243610239775947438443145816369014190026199265338477}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{17} - \frac{7394756490180674542769217761530193617205756622225220301239235983197715281177845301671567002891453208784859857020354531238395954}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{16} + \frac{41300758082079679232457535798810693483835649886188620919505319736039900764528858423377475618924565333900777478428486373068837640}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{15} - \frac{48662112451955252820122068712529864460018301357278201129851754142468674349993965674055652728069796991796019691863085765403278}{2654010107081509076682678664169221742403358094952088092044259545911752406566873979893844250622690299549299019713293338669} a^{14} + \frac{44809501158569469032795930516654379396158409237548404668618162425696049227411772586277098831783966101561650017187642804221081354}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{13} - \frac{9493686369000987151138482244425891646773497332099756089328724898401825180116469537321104455383784096167858050583140049725914026}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{12} + \frac{35377788232033544814644798894902593797617645194678212529421422970748307625884388420804786323895886716040369476057286029849009047}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{11} - \frac{8219108550334759634953045518098346578913616064979334572748385896400953418967344138879599683400717577980275499860884389541601848}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{10} + \frac{19437032457038202364104100971669649395124528327493531560356875595624253384177717454236472685144526535538888786685029307280896215}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{9} - \frac{4440000436219492615328788387842532698046190713111590593095898334062141003747029849203187520927559138792010768786816756654125109}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{8} + \frac{7439549566205786012486941680836316500916702844649743722156437410181980565438755455536776729204497547291554875603309289526443159}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{7} - \frac{1650286844847391297226801540874103183319642151784267160172333161649576470867587076909053165406358276209808477769100329822423403}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{6} + \frac{1630954284227214635222544077750353107363956169328809736348105137552427626454548062013816911629820960763273963732114574093608528}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{5} - \frac{18865115338937451067497325038088670273689584175452585484255794625850060871058870351649043500880234360389801308385629902551582}{48013455573565482387259368560879556976206205535951411846982513603312611718800720181715909624901397237300954992995034035921} a^{4} + \frac{174721050270649397259856360583220809262894721711803963656259622088751190043773710521583613542197165890860348964023173236392147}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a^{3} - \frac{52553416832376270021594965569379671772587494941620035657270408271068722231279575469016765740020162454338254687172243197187429}{1056296022618440612519706108339350253476536521790931060633615299272877457813615843997750011747830739220621009845890748790262} a^{2} + \frac{9359544476860310013598284794086440623991932150558729809753900766017344287914648462506581278241441845116691602118260780680756}{528148011309220306259853054169675126738268260895465530316807649636438728906807921998875005873915369610310504922945374395131} a - \frac{800554190107225009256403417362448574146313458685841200149207384150587200414525008802459068819223624420971712712912575579}{5934247318081127036627562406400844120654699560623208205806827524004929538278740696616573099706914265284387695763431172979} \)
(order $6$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
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 $
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
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 | R | ${\href{/padicField/7.10.0.1}{10} }^{4}$ | R | ${\href{/padicField/13.10.0.1}{10} }^{4}$ | ${\href{/padicField/17.10.0.1}{10} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{20}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{8}$ | ${\href{/padicField/37.10.0.1}{10} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{20}$ | ${\href{/padicField/47.10.0.1}{10} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $2$ | $10$ | $30$ | |||
| Deg $20$ | $2$ | $10$ | $30$ | ||||
|
\(3\)
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ |
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ | |
|
\(5\)
| Deg $20$ | $2$ | $10$ | $10$ | |||
| Deg $20$ | $2$ | $10$ | $10$ | ||||
|
\(11\)
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |