↑

Properties

Label 32.0.61303936503...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{88}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $59.51$
Ramified primes $2, 5, 7$
Class number $680$ (GRH)
Class group $[680]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![38416, -153664, 285376, -219520, -297528, 838880, -340368, -1826272, 4403596, -3851152, -794184, 5679776, -4743542, -3905608, 12201456, -12655248, 5433691, 1957436, -2324538, 543956, -43153, 117740, 1408, -51984, 18183, 2488, -1364, -84, -13, 60, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^31 - 6*x^30 + 60*x^29 - 13*x^28 - 84*x^27 - 1364*x^26 + 2488*x^25 + 18183*x^24 - 51984*x^23 + 1408*x^22 + 117740*x^21 - 43153*x^20 + 543956*x^19 - 2324538*x^18 + 1957436*x^17 + 5433691*x^16 - 12655248*x^15 + 12201456*x^14 - 3905608*x^13 - 4743542*x^12 + 5679776*x^11 - 794184*x^10 - 3851152*x^9 + 4403596*x^8 - 1826272*x^7 - 340368*x^6 + 838880*x^5 - 297528*x^4 - 219520*x^3 + 285376*x^2 - 153664*x + 38416)
 
gp: K = bnfinit(x^32 - 4*x^31 - 6*x^30 + 60*x^29 - 13*x^28 - 84*x^27 - 1364*x^26 + 2488*x^25 + 18183*x^24 - 51984*x^23 + 1408*x^22 + 117740*x^21 - 43153*x^20 + 543956*x^19 - 2324538*x^18 + 1957436*x^17 + 5433691*x^16 - 12655248*x^15 + 12201456*x^14 - 3905608*x^13 - 4743542*x^12 + 5679776*x^11 - 794184*x^10 - 3851152*x^9 + 4403596*x^8 - 1826272*x^7 - 340368*x^6 + 838880*x^5 - 297528*x^4 - 219520*x^3 + 285376*x^2 - 153664*x + 38416, 1)
 

Normalized defining polynomial

\( x^{32} - 4 x^{31} - 6 x^{30} + 60 x^{29} - 13 x^{28} - 84 x^{27} - 1364 x^{26} + 2488 x^{25} + 18183 x^{24} - 51984 x^{23} + 1408 x^{22} + 117740 x^{21} - 43153 x^{20} + 543956 x^{19} - 2324538 x^{18} + 1957436 x^{17} + 5433691 x^{16} - 12655248 x^{15} + 12201456 x^{14} - 3905608 x^{13} - 4743542 x^{12} + 5679776 x^{11} - 794184 x^{10} - 3851152 x^{9} + 4403596 x^{8} - 1826272 x^{7} - 340368 x^{6} + 838880 x^{5} - 297528 x^{4} - 219520 x^{3} + 285376 x^{2} - 153664 x + 38416 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(613039365036788240314949190025216000000000000000000000000=2^{88}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.51$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(560=2^{4}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{560}(1,·)$, $\chi_{560}(363,·)$, $\chi_{560}(321,·)$, $\chi_{560}(393,·)$, $\chi_{560}(267,·)$, $\chi_{560}(531,·)$, $\chi_{560}(281,·)$, $\chi_{560}(153,·)$, $\chi_{560}(27,·)$, $\chi_{560}(547,·)$, $\chi_{560}(41,·)$, $\chi_{560}(43,·)$, $\chi_{560}(433,·)$, $\chi_{560}(307,·)$, $\chi_{560}(57,·)$, $\chi_{560}(449,·)$, $\chi_{560}(323,·)$, $\chi_{560}(211,·)$, $\chi_{560}(337,·)$, $\chi_{560}(419,·)$, $\chi_{560}(139,·)$, $\chi_{560}(97,·)$, $\chi_{560}(99,·)$, $\chi_{560}(209,·)$, $\chi_{560}(379,·)$, $\chi_{560}(489,·)$, $\chi_{560}(491,·)$, $\chi_{560}(113,·)$, $\chi_{560}(83,·)$, $\chi_{560}(169,·)$, $\chi_{560}(377,·)$, $\chi_{560}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{142} a^{22} + \frac{21}{142} a^{21} + \frac{17}{71} a^{20} - \frac{14}{71} a^{19} - \frac{47}{142} a^{18} - \frac{37}{142} a^{17} + \frac{16}{71} a^{16} + \frac{31}{71} a^{15} - \frac{35}{142} a^{14} - \frac{1}{142} a^{13} + \frac{16}{71} a^{12} - \frac{12}{71} a^{11} - \frac{33}{142} a^{10} - \frac{31}{142} a^{9} - \frac{11}{71} a^{8} - \frac{13}{71} a^{7} + \frac{43}{142} a^{6} - \frac{49}{142} a^{5} + \frac{30}{71} a^{4} - \frac{14}{71} a^{3} + \frac{19}{71} a^{2} - \frac{17}{71} a + \frac{3}{71}$, $\frac{1}{142} a^{23} + \frac{19}{142} a^{21} - \frac{16}{71} a^{20} - \frac{27}{142} a^{19} - \frac{22}{71} a^{18} - \frac{43}{142} a^{17} - \frac{21}{71} a^{16} - \frac{59}{142} a^{15} + \frac{12}{71} a^{14} + \frac{53}{142} a^{13} + \frac{7}{71} a^{12} + \frac{45}{142} a^{11} - \frac{24}{71} a^{10} + \frac{61}{142} a^{9} + \frac{5}{71} a^{8} + \frac{21}{142} a^{7} + \frac{21}{71} a^{6} - \frac{47}{142} a^{5} - \frac{5}{71} a^{4} + \frac{29}{71} a^{3} + \frac{10}{71} a^{2} + \frac{5}{71} a + \frac{8}{71}$, $\frac{1}{284} a^{24} + \frac{33}{142} a^{21} + \frac{37}{284} a^{20} - \frac{20}{71} a^{19} - \frac{1}{142} a^{18} + \frac{11}{142} a^{17} - \frac{99}{284} a^{16} + \frac{31}{71} a^{15} - \frac{67}{142} a^{14} - \frac{19}{142} a^{13} + \frac{5}{284} a^{12} + \frac{31}{71} a^{11} + \frac{30}{71} a^{10} + \frac{51}{142} a^{9} + \frac{13}{284} a^{8} - \frac{8}{71} a^{7} + \frac{65}{142} a^{6} - \frac{1}{142} a^{5} - \frac{22}{71} a^{4} - \frac{4}{71} a^{3} + \frac{35}{71} a^{2} - \frac{12}{71} a + \frac{7}{71}$, $\frac{1}{284} a^{25} - \frac{1}{4} a^{21} - \frac{13}{71} a^{20} - \frac{1}{2} a^{19} - \frac{1}{4} a^{17} + \frac{17}{142} a^{15} - \frac{1}{4} a^{13} + \frac{2}{71} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{30}{71} a^{5} - \frac{28}{71}$, $\frac{1}{5964} a^{26} + \frac{5}{2982} a^{25} + \frac{1}{5964} a^{24} + \frac{1}{1491} a^{23} - \frac{13}{5964} a^{22} + \frac{21}{142} a^{21} + \frac{1219}{5964} a^{20} - \frac{169}{1491} a^{19} + \frac{1285}{5964} a^{18} - \frac{1219}{2982} a^{17} + \frac{2759}{5964} a^{16} + \frac{25}{213} a^{15} - \frac{2707}{5964} a^{14} + \frac{67}{142} a^{13} - \frac{41}{84} a^{12} - \frac{90}{497} a^{11} + \frac{11}{5964} a^{10} + \frac{481}{2982} a^{9} - \frac{797}{5964} a^{8} + \frac{19}{213} a^{7} + \frac{100}{1491} a^{6} + \frac{26}{497} a^{5} + \frac{625}{1491} a^{4} + \frac{713}{1491} a^{3} - \frac{11}{497} a^{2} - \frac{8}{71} a + \frac{67}{213}$, $\frac{1}{5964} a^{27} + \frac{1}{994} a^{25} - \frac{1}{994} a^{24} - \frac{11}{5964} a^{23} + \frac{1}{1491} a^{22} + \frac{179}{2982} a^{21} - \frac{19}{426} a^{20} + \frac{2333}{5964} a^{19} + \frac{5}{71} a^{18} + \frac{107}{426} a^{17} - \frac{635}{2982} a^{16} + \frac{457}{5964} a^{15} + \frac{2}{21} a^{14} + \frac{128}{1491} a^{13} + \frac{1163}{2982} a^{12} + \frac{1109}{5964} a^{11} - \frac{237}{497} a^{10} - \frac{977}{2982} a^{9} + \frac{3}{14} a^{8} + \frac{108}{497} a^{7} + \frac{611}{1491} a^{6} - \frac{793}{2982} a^{5} + \frac{16}{213} a^{4} + \frac{502}{1491} a^{3} - \frac{86}{497} a^{2} + \frac{55}{213} a - \frac{4}{213}$, $\frac{1}{2588376} a^{28} + \frac{17}{323547} a^{27} - \frac{11}{215698} a^{26} + \frac{87}{431396} a^{25} - \frac{293}{2588376} a^{24} - \frac{101}{46221} a^{23} + \frac{3259}{1294188} a^{22} + \frac{305947}{1294188} a^{21} + \frac{164877}{862792} a^{20} - \frac{139925}{323547} a^{19} - \frac{562411}{1294188} a^{18} + \frac{4091}{8804} a^{17} - \frac{308011}{862792} a^{16} - \frac{20239}{92442} a^{15} - \frac{53524}{107849} a^{14} - \frac{145689}{431396} a^{13} + \frac{59989}{862792} a^{12} + \frac{113525}{647094} a^{11} + \frac{46141}{1294188} a^{10} + \frac{56053}{184884} a^{9} - \frac{63505}{431396} a^{8} + \frac{317087}{647094} a^{7} - \frac{6218}{107849} a^{6} - \frac{111038}{323547} a^{5} + \frac{97882}{323547} a^{4} - \frac{10783}{46221} a^{3} - \frac{661}{46221} a^{2} - \frac{353}{2201} a - \frac{2741}{6603}$, $\frac{1}{25892851807959760488446047371864} a^{29} - \frac{34763892617096886883653}{616496471618089535439191604092} a^{28} - \frac{76087022356484125286959429}{3236606475994970061055755921483} a^{27} - \frac{937308947914512752820484589}{12946425903979880244223023685932} a^{26} - \frac{13271876804224320954078564303}{8630950602653253496148682457288} a^{25} + \frac{592628198484565219903056105}{4315475301326626748074341228644} a^{24} - \frac{21298946295281758768574454499}{12946425903979880244223023685932} a^{23} + \frac{141039481264089380689375995}{88070924516869933634170229156} a^{22} + \frac{1783493429601154567741901865013}{8630950602653253496148682457288} a^{21} - \frac{6798323287810888451021072341}{616496471618089535439191604092} a^{20} + \frac{8752603191744690669043729319}{264212773550609800902510687468} a^{19} + \frac{3742736564547532137886496720351}{12946425903979880244223023685932} a^{18} - \frac{2562779289377948987065696349105}{25892851807959760488446047371864} a^{17} + \frac{5810066290871949466779732123101}{12946425903979880244223023685932} a^{16} + \frac{225982863506937251431989538499}{6473212951989940122111511842966} a^{15} - \frac{596010256046710997070240957465}{4315475301326626748074341228644} a^{14} - \frac{4012409281140947280475007254433}{25892851807959760488446047371864} a^{13} + \frac{264056201179756993185972520537}{12946425903979880244223023685932} a^{12} - \frac{1204030673874798444505423500205}{12946425903979880244223023685932} a^{11} - \frac{3758067219936248294184445455439}{12946425903979880244223023685932} a^{10} - \frac{230876868574112271867235306253}{4315475301326626748074341228644} a^{9} + \frac{639692519516549283526753477993}{6473212951989940122111511842966} a^{8} - \frac{209380695800573373739542371057}{1078868825331656687018585307161} a^{7} + \frac{210328639460104909404342020573}{924744707427134303158787406138} a^{6} - \frac{689507862678328687091270545661}{3236606475994970061055755921483} a^{5} - \frac{424548965134666307289274578274}{1078868825331656687018585307161} a^{4} + \frac{56177681770775890814507296008}{154124117904522383859797901023} a^{3} - \frac{181043617723176164412793755764}{462372353713567151579393703069} a^{2} + \frac{25161684556379734326759967340}{66053193387652450225627671867} a + \frac{8835815773993809201242594972}{66053193387652450225627671867}$, $\frac{1}{12868747348556000962757685543816408} a^{30} - \frac{151}{12868747348556000962757685543816408} a^{29} + \frac{2348460039913900136567871823}{12868747348556000962757685543816408} a^{28} - \frac{387830527227144020769956761049}{6434373674278000481378842771908204} a^{27} + \frac{893240446275732989117990301703}{12868747348556000962757685543816408} a^{26} + \frac{210408254183411356150161253193}{612797492788380998226556454467448} a^{25} + \frac{4461744909426002638026532355803}{12868747348556000962757685543816408} a^{24} - \frac{1346933646460258571361093246469}{1072395612379666746896473795318034} a^{23} - \frac{24199727700680647473570525184259}{12868747348556000962757685543816408} a^{22} + \frac{1821561333666097643852848508041441}{12868747348556000962757685543816408} a^{21} - \frac{245165780104486530325408748198605}{12868747348556000962757685543816408} a^{20} + \frac{212715671406796669994291005878191}{459598119591285748669917340850586} a^{19} + \frac{962461172305790884134376069536375}{4289582449518666987585895181272136} a^{18} + \frac{128028654295985175882524757132535}{1838392478365142994679669363402344} a^{17} - \frac{1125130643678979999285797380092665}{4289582449518666987585895181272136} a^{16} - \frac{95059614129801766083772069558469}{6434373674278000481378842771908204} a^{15} - \frac{155200636446479140398781201651681}{12868747348556000962757685543816408} a^{14} + \frac{622452154413902952578040461196891}{4289582449518666987585895181272136} a^{13} - \frac{693126517768684216151335758157327}{4289582449518666987585895181272136} a^{12} - \frac{180011700445300351946473727974393}{459598119591285748669917340850586} a^{11} - \frac{1087632201829007765416162729217527}{3217186837139000240689421385954102} a^{10} + \frac{155868193144591173545126942573514}{536197806189833373448236897659017} a^{9} + \frac{231742404702038096396364042615187}{6434373674278000481378842771908204} a^{8} + \frac{1360967426857947049302131331367061}{3217186837139000240689421385954102} a^{7} + \frac{438922015677217049159727716147299}{1608593418569500120344710692977051} a^{6} + \frac{70825723201618244368568481135911}{459598119591285748669917340850586} a^{5} + \frac{148771333686013356912150209197165}{459598119591285748669917340850586} a^{4} - \frac{5364468659051952272806891810443}{10942812371221089254045650972633} a^{3} + \frac{459061016977101976608224485489}{4689776730523323966019564702557} a^{2} + \frac{1265463568579870149830459553152}{4689776730523323966019564702557} a + \frac{2294045508920470216656393986765}{4689776730523323966019564702557}$, $\frac{1}{12868747348556000962757685543816408} a^{31} + \frac{1}{207560441105741951012220734577684} a^{29} - \frac{266730542436887404664795887}{12868747348556000962757685543816408} a^{28} - \frac{216057785848679295396495758515}{4289582449518666987585895181272136} a^{27} + \frac{60443150571729759310432903439}{6434373674278000481378842771908204} a^{26} + \frac{341454714461463588276805509847}{536197806189833373448236897659017} a^{25} + \frac{2390534338291304375559351248591}{1838392478365142994679669363402344} a^{24} - \frac{38569024929284221377209277979813}{12868747348556000962757685543816408} a^{23} + \frac{78273405492441607562887214429}{153199373197095249556639113616862} a^{22} - \frac{9668074259229247215743330819435}{153199373197095249556639113616862} a^{21} + \frac{2362475119516580402935894152241589}{12868747348556000962757685543816408} a^{20} - \frac{2010446400607675256113405600670033}{12868747348556000962757685543816408} a^{19} + \frac{481638637970289634585063513508477}{1608593418569500120344710692977051} a^{18} - \frac{2386866135605295969970326798345637}{6434373674278000481378842771908204} a^{17} - \frac{1737551667055258174030309868609865}{4289582449518666987585895181272136} a^{16} - \frac{2228361023069972351306545616849393}{12868747348556000962757685543816408} a^{15} + \frac{39972332660794757035730393625823}{207560441105741951012220734577684} a^{14} - \frac{258709775593113512806392421158493}{536197806189833373448236897659017} a^{13} + \frac{853637867659066216970263281852135}{4289582449518666987585895181272136} a^{12} + \frac{2396291811725989843814016660582025}{6434373674278000481378842771908204} a^{11} - \frac{462101370205951757561171616031605}{1072395612379666746896473795318034} a^{10} + \frac{153934005066246135062084557578490}{1608593418569500120344710692977051} a^{9} - \frac{2341463160049826631561242252735}{7412872896633641107579311949203} a^{8} - \frac{558690506713129607704839623345291}{3217186837139000240689421385954102} a^{7} + \frac{28218392586004978681958020003121}{3217186837139000240689421385954102} a^{6} - \frac{174063025432151523147938818782353}{459598119591285748669917340850586} a^{5} - \frac{3057712826549937125487837675768}{10942812371221089254045650972633} a^{4} + \frac{15859085489313431365377940945748}{32828437113663267762136952917899} a^{3} - \frac{3406435334669600618586969918610}{10942812371221089254045650972633} a^{2} + \frac{611380698728315391779152296781}{1563258910174441322006521567519} a + \frac{274741505500424805207362130203}{4689776730523323966019564702557}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{680}$, which has order $680$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{2949627240068037965455608208}{51890110276435487753055183644421} a^{31} + \frac{7149781932162916432429913271}{34593406850956991835370122429614} a^{30} + \frac{3025078347277826118120423089}{7412872896633641107579311949203} a^{29} - \frac{673229767406774154048383684155}{207560441105741951012220734577684} a^{28} - \frac{17880476874325997378726752040}{51890110276435487753055183644421} a^{27} + \frac{443987460092834894203648040071}{103780220552870975506110367288842} a^{26} + \frac{4068339914516917686877847515615}{51890110276435487753055183644421} a^{25} - \frac{1946999117674670917820122435466}{17296703425478495917685061214807} a^{24} - \frac{55115211738314466869573054444747}{51890110276435487753055183644421} a^{23} + \frac{88844646610043086562256885264177}{34593406850956991835370122429614} a^{22} + \frac{35431008656281615918702800147913}{51890110276435487753055183644421} a^{21} - \frac{326843890675303786113650640184664}{51890110276435487753055183644421} a^{20} + \frac{37203529655094829404651810599054}{51890110276435487753055183644421} a^{19} - \frac{1618533640040639718344615676623224}{51890110276435487753055183644421} a^{18} + \frac{2067670680049087680696722234389665}{17296703425478495917685061214807} a^{17} - \frac{539259814858414560545146595849963}{7412872896633641107579311949203} a^{16} - \frac{800284744601782481195886952938364}{2470957632211213702526437316401} a^{15} + \frac{63696306398216291838408478925329663}{103780220552870975506110367288842} a^{14} - \frac{3870390882544480697259027796194940}{7412872896633641107579311949203} a^{13} + \frac{763445225294806898976157614987904}{17296703425478495917685061214807} a^{12} + \frac{16119882028554042784369238374787960}{51890110276435487753055183644421} a^{11} - \frac{12590852591249060352881799690272287}{51890110276435487753055183644421} a^{10} - \frac{608582913386531623357641565246916}{17296703425478495917685061214807} a^{9} + \frac{15928184663448552320398391681550897}{69186813701913983670740244859228} a^{8} - \frac{1379534272198924057837735658766112}{7412872896633641107579311949203} a^{7} + \frac{2020340765522118996705008559400142}{51890110276435487753055183644421} a^{6} + \frac{45835507302488488071809680238312}{1058981842376234443939901707029} a^{5} - \frac{301944471126697677926331135926584}{7412872896633641107579311949203} a^{4} - \frac{80463447187134676334297469656}{151283120339462063419985958147} a^{3} + \frac{5627671971239143201458403494300}{352993947458744814646633902343} a^{2} - \frac{629113857199261291167818159120}{50427706779820687806661986049} a + \frac{672950835557919872315719962592}{151283120339462063419985958147} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 14712129696811.27 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}, \sqrt{-35})\), 4.4.2508800.1, 4.0.2048.2, \(\Q(\sqrt{10}, \sqrt{-14})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-7})\), 4.4.100352.1, 4.0.51200.2, \(\Q(\sqrt{-7}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-14})\), 4.4.256000.1, 4.0.12544000.2, 4.4.256000.2, 4.0.12544000.1, 4.4.392000.1, 4.0.8000.2, 4.4.6125.1, \(\Q(\zeta_{5})\), 8.0.6294077440000.4, 8.0.6146560000.2, 8.0.6294077440000.1, 8.8.6294077440000.1, 8.0.6294077440000.3, 8.0.2621440000.1, 8.0.10070523904.1, 8.0.157351936000000.68, 8.0.157351936000000.70, 8.0.153664000000.2, 8.0.37515625.1, 8.8.65536000000.1, 8.0.157351936000000.83, 8.8.153664000000.1, 8.0.64000000.2, 8.0.157351936000000.31, 8.0.157351936000000.65, 8.0.153664000000.5, 8.0.153664000000.1, 16.0.39615410820716953600000000.2, 16.0.24759631762948096000000000000.1, 16.0.23612624896000000000000.2, 16.16.24759631762948096000000000000.1, 16.0.24759631762948096000000000000.5, 16.0.4294967296000000000000.2, 16.0.24759631762948096000000000000.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{8}$ R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
5Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$