Properties

Label 32.0.133...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.335\times 10^{50}$
Root discriminant $36.85$
Ramified primes $2, 3, 5, 13, 97$
Class number $20$ (GRH)
Class group $[20]$ (GRH)
Galois group 32T12882

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 16*x^30 + 24*x^29 + 141*x^28 - 120*x^27 - 852*x^26 - 70*x^25 + 4216*x^24 + 4978*x^23 - 15640*x^22 - 47344*x^21 + 45797*x^20 + 270436*x^19 - 110380*x^18 - 986938*x^17 + 284105*x^16 + 1973876*x^15 - 441520*x^14 - 2163488*x^13 + 732752*x^12 + 1515008*x^11 - 1000960*x^10 - 637184*x^9 + 1079296*x^8 + 35840*x^7 - 872448*x^6 + 245760*x^5 + 577536*x^4 - 196608*x^3 - 262144*x^2 + 65536*x + 65536)
 
gp: K = bnfinit(x^32 - 2*x^31 - 16*x^30 + 24*x^29 + 141*x^28 - 120*x^27 - 852*x^26 - 70*x^25 + 4216*x^24 + 4978*x^23 - 15640*x^22 - 47344*x^21 + 45797*x^20 + 270436*x^19 - 110380*x^18 - 986938*x^17 + 284105*x^16 + 1973876*x^15 - 441520*x^14 - 2163488*x^13 + 732752*x^12 + 1515008*x^11 - 1000960*x^10 - 637184*x^9 + 1079296*x^8 + 35840*x^7 - 872448*x^6 + 245760*x^5 + 577536*x^4 - 196608*x^3 - 262144*x^2 + 65536*x + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 65536, -262144, -196608, 577536, 245760, -872448, 35840, 1079296, -637184, -1000960, 1515008, 732752, -2163488, -441520, 1973876, 284105, -986938, -110380, 270436, 45797, -47344, -15640, 4978, 4216, -70, -852, -120, 141, 24, -16, -2, 1]);
 

\( x^{32} - 2 x^{31} - 16 x^{30} + 24 x^{29} + 141 x^{28} - 120 x^{27} - 852 x^{26} - 70 x^{25} + 4216 x^{24} + 4978 x^{23} - 15640 x^{22} - 47344 x^{21} + 45797 x^{20} + 270436 x^{19} - 110380 x^{18} - 986938 x^{17} + 284105 x^{16} + 1973876 x^{15} - 441520 x^{14} - 2163488 x^{13} + 732752 x^{12} + 1515008 x^{11} - 1000960 x^{10} - 637184 x^{9} + 1079296 x^{8} + 35840 x^{7} - 872448 x^{6} + 245760 x^{5} + 577536 x^{4} - 196608 x^{3} - 262144 x^{2} + 65536 x + 65536 \)

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(133516116647406931991162570051420160000000000000000\)\(\medspace = 2^{48}\cdot 3^{16}\cdot 5^{16}\cdot 13^{8}\cdot 97^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $36.85$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 5, 13, 97$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{5} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{6} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{7} + \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{8} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{17} - \frac{1}{8} a^{16} - \frac{1}{8} a^{15} - \frac{1}{4} a^{13} + \frac{1}{4} a^{12} - \frac{1}{2} a^{11} + \frac{1}{4} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{8} a^{6} + \frac{3}{8} a^{5} + \frac{3}{8} a^{4} + \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{19} - \frac{1}{8} a^{15} - \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{3}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{20} + \frac{1}{16} a^{16} - \frac{1}{8} a^{15} + \frac{1}{8} a^{13} + \frac{1}{4} a^{12} - \frac{3}{8} a^{11} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{5}{16} a^{8} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{8} a^{5} - \frac{7}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{21} - \frac{3}{32} a^{17} + \frac{1}{16} a^{16} + \frac{1}{16} a^{14} - \frac{1}{8} a^{13} + \frac{5}{16} a^{12} + \frac{3}{8} a^{11} + \frac{1}{4} a^{10} + \frac{5}{32} a^{9} + \frac{7}{16} a^{8} + \frac{7}{16} a^{6} + \frac{5}{32} a^{5} - \frac{1}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{22} - \frac{3}{64} a^{18} + \frac{1}{32} a^{17} + \frac{1}{32} a^{15} - \frac{1}{16} a^{14} + \frac{5}{32} a^{13} + \frac{3}{16} a^{12} + \frac{1}{8} a^{11} + \frac{5}{64} a^{10} - \frac{9}{32} a^{9} - \frac{9}{32} a^{7} + \frac{5}{64} a^{6} - \frac{1}{32} a^{5} + \frac{3}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{128} a^{23} - \frac{3}{128} a^{19} + \frac{1}{64} a^{18} + \frac{1}{64} a^{16} - \frac{1}{32} a^{15} + \frac{5}{64} a^{14} + \frac{3}{32} a^{13} + \frac{1}{16} a^{12} - \frac{59}{128} a^{11} + \frac{23}{64} a^{10} + \frac{23}{64} a^{8} - \frac{59}{128} a^{7} - \frac{1}{64} a^{6} + \frac{3}{32} a^{5} - \frac{1}{4} a^{4} + \frac{3}{8} a^{3}$, $\frac{1}{768} a^{24} + \frac{1}{384} a^{23} + \frac{1}{192} a^{22} - \frac{19}{768} a^{20} - \frac{1}{192} a^{19} - \frac{1}{96} a^{18} - \frac{43}{384} a^{17} + \frac{1}{16} a^{16} + \frac{37}{384} a^{15} + \frac{1}{48} a^{14} - \frac{11}{96} a^{13} + \frac{133}{768} a^{12} - \frac{41}{96} a^{11} - \frac{5}{48} a^{10} + \frac{115}{384} a^{9} + \frac{91}{256} a^{8} - \frac{1}{24} a^{7} + \frac{55}{192} a^{6} - \frac{13}{48} a^{5} - \frac{19}{48} a^{4} + \frac{1}{8} a^{3} - \frac{5}{12} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{1536} a^{25} - \frac{1}{192} a^{22} - \frac{19}{1536} a^{21} + \frac{17}{768} a^{20} - \frac{35}{768} a^{18} - \frac{41}{384} a^{17} - \frac{11}{768} a^{16} - \frac{11}{128} a^{15} - \frac{5}{64} a^{14} + \frac{103}{512} a^{13} + \frac{29}{256} a^{12} - \frac{1}{8} a^{11} - \frac{63}{256} a^{10} + \frac{581}{1536} a^{9} - \frac{289}{768} a^{8} - \frac{121}{384} a^{7} + \frac{5}{64} a^{6} + \frac{31}{96} a^{5} - \frac{1}{24} a^{4} + \frac{1}{6} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{3072} a^{26} - \frac{1}{384} a^{23} - \frac{19}{3072} a^{22} + \frac{17}{1536} a^{21} - \frac{35}{1536} a^{19} - \frac{41}{768} a^{18} - \frac{11}{1536} a^{17} + \frac{21}{256} a^{16} + \frac{11}{128} a^{15} + \frac{103}{1024} a^{14} - \frac{99}{512} a^{13} + \frac{3}{16} a^{12} - \frac{63}{512} a^{11} + \frac{581}{3072} a^{10} + \frac{479}{1536} a^{9} - \frac{121}{768} a^{8} + \frac{37}{128} a^{7} + \frac{79}{192} a^{6} - \frac{1}{48} a^{5} - \frac{7}{24} a^{4} - \frac{1}{4} a^{2} + \frac{1}{3} a$, $\frac{1}{6144} a^{27} - \frac{1}{2048} a^{23} - \frac{5}{1024} a^{22} + \frac{27}{1024} a^{20} - \frac{49}{1536} a^{19} + \frac{101}{3072} a^{18} - \frac{157}{1536} a^{17} + \frac{11}{256} a^{16} + \frac{709}{6144} a^{15} - \frac{41}{3072} a^{14} + \frac{19}{96} a^{13} + \frac{1303}{3072} a^{12} - \frac{681}{2048} a^{11} + \frac{229}{1024} a^{10} - \frac{255}{512} a^{9} - \frac{3}{16} a^{8} - \frac{55}{128} a^{7} + \frac{43}{96} a^{6} - \frac{1}{96} a^{5} + \frac{17}{48} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{81051648} a^{28} - \frac{2383}{40525824} a^{27} + \frac{491}{10131456} a^{26} + \frac{2869}{10131456} a^{25} + \frac{10559}{27017216} a^{24} - \frac{39089}{20262912} a^{23} - \frac{109073}{20262912} a^{22} + \frac{619693}{40525824} a^{21} - \frac{74795}{5065728} a^{20} - \frac{89927}{40525824} a^{19} + \frac{2753}{5065728} a^{18} + \frac{139373}{1688576} a^{17} - \frac{9262459}{81051648} a^{16} + \frac{106269}{3377152} a^{15} - \frac{138745}{6754304} a^{14} + \frac{563041}{13508608} a^{13} - \frac{14338159}{81051648} a^{12} + \frac{1307137}{3377152} a^{11} - \frac{1692349}{5065728} a^{10} - \frac{2227573}{5065728} a^{9} - \frac{608027}{1266432} a^{8} - \frac{245039}{633216} a^{7} - \frac{45479}{316608} a^{6} + \frac{72257}{158304} a^{5} - \frac{1895}{6596} a^{4} - \frac{11275}{79152} a^{3} - \frac{1019}{19788} a^{2} - \frac{181}{9894} a - \frac{4943}{19788}$, $\frac{1}{162103296} a^{29} + \frac{483}{13508608} a^{27} + \frac{465}{6754304} a^{26} - \frac{6641}{54034432} a^{25} - \frac{16709}{27017216} a^{24} + \frac{44743}{13508608} a^{23} + \frac{137563}{27017216} a^{22} - \frac{547081}{40525824} a^{21} + \frac{52025}{81051648} a^{20} + \frac{2063771}{40525824} a^{19} + \frac{19997}{3377152} a^{18} + \frac{8013733}{162103296} a^{17} + \frac{7283239}{81051648} a^{16} + \frac{710461}{40525824} a^{15} + \frac{3029455}{81051648} a^{14} - \frac{588633}{3178496} a^{13} - \frac{10759767}{27017216} a^{12} + \frac{1627743}{3377152} a^{11} - \frac{870521}{3377152} a^{10} - \frac{102157}{1688576} a^{9} - \frac{122875}{1266432} a^{8} - \frac{22225}{158304} a^{7} + \frac{75265}{158304} a^{6} - \frac{25219}{52768} a^{5} + \frac{10545}{52768} a^{4} + \frac{5039}{79152} a^{3} + \frac{3817}{19788} a^{2} - \frac{2893}{13192} a + \frac{429}{6596}$, $\frac{1}{598402848352660034974740185088} a^{30} + \frac{326776950826175549251}{299201424176330017487370092544} a^{29} + \frac{158218030228950058239}{49866904029388336247895015424} a^{28} + \frac{2969427242425070657756047}{37400178022041252185921261568} a^{27} + \frac{13955385684264217753300925}{598402848352660034974740185088} a^{26} - \frac{1704016182371137552507467}{12466726007347084061973753856} a^{25} - \frac{5388129617577487362777799}{12466726007347084061973753856} a^{24} - \frac{427271228343043661791050467}{299201424176330017487370092544} a^{23} - \frac{157028167863589510945366393}{24933452014694168123947507712} a^{22} - \frac{1921084837330397278289538019}{299201424176330017487370092544} a^{21} - \frac{1093189392261517330167651013}{74800356044082504371842523136} a^{20} + \frac{590087980565331676139811683}{74800356044082504371842523136} a^{19} - \frac{25986524818727712383379542267}{598402848352660034974740185088} a^{18} + \frac{5395426288012479615807019723}{149600712088165008743685046272} a^{17} + \frac{1285530843192322617504396811}{74800356044082504371842523136} a^{16} + \frac{19874165932262520156471595675}{299201424176330017487370092544} a^{15} + \frac{4462854213217792021100121683}{199467616117553344991580061696} a^{14} + \frac{18436073132524687021092301213}{149600712088165008743685046272} a^{13} - \frac{39802758537167495577023850871}{149600712088165008743685046272} a^{12} - \frac{5531388674347704395499175429}{12466726007347084061973753856} a^{11} - \frac{1388786748997080835247866001}{9350044505510313046480315392} a^{10} + \frac{1198630364576102253551123087}{3116681501836771015493438464} a^{9} - \frac{68271250132769506479929095}{146094445398598641351254928} a^{8} + \frac{407002032361370486853134455}{1168755563188789130810039424} a^{7} - \frac{162661519246349830237865695}{584377781594394565405019712} a^{6} + \frac{61532709872252597077400279}{194792593864798188468339904} a^{5} - \frac{30893090842487416852445149}{73047222699299320675627464} a^{4} - \frac{14101993637201260406596607}{146094445398598641351254928} a^{3} - \frac{33118316700284851791528935}{146094445398598641351254928} a^{2} + \frac{3333018920653225974891755}{12174537116549886779271244} a - \frac{312148130864346702907331}{2148447726449980019871396}$, $\frac{1}{12434334860100986809387261152941309952} a^{31} + \frac{73171}{777145928756311675586703822058831872} a^{30} + \frac{29477130191122850739523505}{45714466397430098563923754238754816} a^{29} - \frac{6824284370529659194782074789}{1554291857512623351173407644117663744} a^{28} - \frac{967674834811546390325602135249187}{12434334860100986809387261152941309952} a^{27} - \frac{320557083149055446369122210179309}{2072389143350164468231210192156884992} a^{26} + \frac{63615569273365910854525357213849}{259048642918770558528901274019610624} a^{25} + \frac{881689787942793104202308654813343}{2072389143350164468231210192156884992} a^{24} + \frac{4433501758661806296808687266301931}{3108583715025246702346815288235327488} a^{23} - \frac{3479216098785400890778815642709481}{2072389143350164468231210192156884992} a^{22} + \frac{39707675432421183825714734644184807}{3108583715025246702346815288235327488} a^{21} - \frac{43468549124274763282533561858252679}{1554291857512623351173407644117663744} a^{20} - \frac{267070758095842630687405346494535627}{12434334860100986809387261152941309952} a^{19} + \frac{211274765151909098065987395850807231}{6217167430050493404693630576470654976} a^{18} - \frac{6468100326318932274876305609802575}{259048642918770558528901274019610624} a^{17} + \frac{84985746751804137246553491743520611}{6217167430050493404693630576470654976} a^{16} - \frac{8242218207398968603693269832081019}{731431462358881577022780067820077056} a^{15} + \frac{256988163054841813172219564249679405}{2072389143350164468231210192156884992} a^{14} + \frac{758477515494939438046406343507817283}{3108583715025246702346815288235327488} a^{13} + \frac{569596929385992126957412021220503507}{1554291857512623351173407644117663744} a^{12} + \frac{2055139382673627499520922727675053}{7619077732905016427320625706459136} a^{11} - \frac{46185784680388740482242192644997253}{194286482189077918896675955514707968} a^{10} + \frac{11307952817233586900611399102397533}{32381080364846319816112659252451328} a^{9} + \frac{4586730622439270414920686862143763}{12142905136817369931042247219669248} a^{8} - \frac{1719254338278055710461972536704665}{6071452568408684965521123609834624} a^{7} - \frac{902720704492500713063455530042323}{4047635045605789977014082406556416} a^{6} + \frac{2377595239436245230780419243143549}{6071452568408684965521123609834624} a^{5} + \frac{1074150274413796427738442133586641}{3035726284204342482760561804917312} a^{4} - \frac{18390655613899566595244306335043}{1011908761401447494253520601639104} a^{3} - \frac{27828908776553173167465074004949}{1517863142102171241380280902458656} a^{2} + \frac{18956989136110083353868247209851}{758931571051085620690140451229328} a + \frac{1543893723390298823524563826061}{7159731802368732270661702370088}$

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

Class group and class number

$C_{20}$, which has order $20$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{6280699354319001510588492377}{126794282073490438265545608134656} a^{31} - \frac{9666981736606408222611246545}{95095711555117828699159206100992} a^{30} - \frac{792684464762217849236039051}{932310897599194399011364765696} a^{29} + \frac{3776509980154824485176227601}{2796932692797583197034094297088} a^{28} + \frac{991231811694610691032390474229}{126794282073490438265545608134656} a^{27} - \frac{1419743233517758053587809909967}{190191423110235657398318412201984} a^{26} - \frac{389217338599865810255463165941}{7924642629593152391596600508416} a^{25} + \frac{163873099324473363393333622925}{63397141036745219132772804067328} a^{24} + \frac{23626843745437979365889952194855}{95095711555117828699159206100992} a^{23} + \frac{16379170780041305952300553922989}{63397141036745219132772804067328} a^{22} - \frac{30643988209742607661689418981999}{31698570518372609566386402033664} a^{21} - \frac{125000682581242613497601485871161}{47547855777558914349579603050496} a^{20} + \frac{1113020852071040759576079056392967}{380382846220471314796636824403968} a^{19} + \frac{3005008846211023299312023226307939}{190191423110235657398318412201984} a^{18} - \frac{28810609678747878190473689318623}{3962321314796576195798300254208} a^{17} - \frac{11577031543418779028247072928613735}{190191423110235657398318412201984} a^{16} + \frac{6659902887413078207172578036770159}{380382846220471314796636824403968} a^{15} + \frac{25268200394982357586660559524821971}{190191423110235657398318412201984} a^{14} - \frac{2623129379045932299512595663183163}{95095711555117828699159206100992} a^{13} - \frac{2271658881534620108179115289352629}{15849285259186304783193201016832} a^{12} + \frac{2033582569420434774760692371797}{74760779524463701807515099136} a^{11} + \frac{464448343730734343162312167692421}{5943481972194864293697450381312} a^{10} - \frac{59214493029518640273303198564893}{990580328699144048949575063552} a^{9} - \frac{255403408873581749831009551823}{7283678887493706242276287232} a^{8} + \frac{87795450643314410778669672941}{1451045403367886790453479097} a^{7} - \frac{844333857272749798470419049659}{371467623262179018356090648832} a^{6} - \frac{9400875130023257148110769082211}{185733811631089509178045324416} a^{5} + \frac{417484008346877247147965935805}{30955635271848251529674220736} a^{4} + \frac{2848138562285835296864712607325}{92866905815544754589022662208} a^{3} - \frac{475593885459481042048592501521}{46433452907772377294511331104} a^{2} - \frac{346738523038436140883932360955}{23216726453886188647255665552} a - \frac{6306616162422131411344170487}{3869454408981031441209277592} \) (order $12$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 564327334704.2394 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 564327334704.2394 \cdot 20}{12\sqrt{133516116647406931991162570051420160000000000000000}}\approx 0.480275999186225$ (assuming GRH)

Galois group

32T12882:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 512
The 80 conjugacy class representatives for t32n12882 are not computed
Character table for t32n12882 is not computed

Intermediate fields

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), 4.4.7488.1, 4.0.7488.1, 4.4.187200.1, 4.0.187200.1, \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{-5})\), Deg 8, Deg 8, 8.0.5438803968.1, 8.8.5438803968.1, 8.0.12960000.1, 8.0.56070144.2, 8.0.35043840000.18, 8.8.35043840000.1, 8.0.35043840000.41, 8.0.35043840000.39, 8.0.35043840000.36, 16.0.1228070721945600000000.2, Deg 16, 16.0.29580588602332545024.3, Deg 16, Deg 16, Deg 16, 16.0.11554917422786150400000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 32 siblings: data not computed

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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.14$x^{8} + 12 x^{4} + 144$$4$$2$$12$$D_4$$[2, 2]^{2}$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5Data not computed
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$97$97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$