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Properties

Label 32.0.24000959919...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{48}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $43.34$
Ramified primes $2, 3, 5, 7$
Class number Not computed
Class group Not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -1024, 1152, 3072, -13184, 24320, -12128, -70592, 240720, -329952, -39944, 1256816, -3068744, 3448824, 2519526, -12810752, 14846451, -3485698, 2319339, -694266, 39466, 57880, -30305, 21504, -7635, 2264, -323, 64, 46, -30, 9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 + 9*x^30 - 30*x^29 + 46*x^28 + 64*x^27 - 323*x^26 + 2264*x^25 - 7635*x^24 + 21504*x^23 - 30305*x^22 + 57880*x^21 + 39466*x^20 - 694266*x^19 + 2319339*x^18 - 3485698*x^17 + 14846451*x^16 - 12810752*x^15 + 2519526*x^14 + 3448824*x^13 - 3068744*x^12 + 1256816*x^11 - 39944*x^10 - 329952*x^9 + 240720*x^8 - 70592*x^7 - 12128*x^6 + 24320*x^5 - 13184*x^4 + 3072*x^3 + 1152*x^2 - 1024*x + 256)
 
gp: K = bnfinit(x^32 - 2*x^31 + 9*x^30 - 30*x^29 + 46*x^28 + 64*x^27 - 323*x^26 + 2264*x^25 - 7635*x^24 + 21504*x^23 - 30305*x^22 + 57880*x^21 + 39466*x^20 - 694266*x^19 + 2319339*x^18 - 3485698*x^17 + 14846451*x^16 - 12810752*x^15 + 2519526*x^14 + 3448824*x^13 - 3068744*x^12 + 1256816*x^11 - 39944*x^10 - 329952*x^9 + 240720*x^8 - 70592*x^7 - 12128*x^6 + 24320*x^5 - 13184*x^4 + 3072*x^3 + 1152*x^2 - 1024*x + 256, 1)
 

Normalized defining polynomial

\( x^{32} - 2 x^{31} + 9 x^{30} - 30 x^{29} + 46 x^{28} + 64 x^{27} - 323 x^{26} + 2264 x^{25} - 7635 x^{24} + 21504 x^{23} - 30305 x^{22} + 57880 x^{21} + 39466 x^{20} - 694266 x^{19} + 2319339 x^{18} - 3485698 x^{17} + 14846451 x^{16} - 12810752 x^{15} + 2519526 x^{14} + 3448824 x^{13} - 3068744 x^{12} + 1256816 x^{11} - 39944 x^{10} - 329952 x^{9} + 240720 x^{8} - 70592 x^{7} - 12128 x^{6} + 24320 x^{5} - 13184 x^{4} + 3072 x^{3} + 1152 x^{2} - 1024 x + 256 \)

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:  \(24000959919026880122072334336000000000000000000000000=2^{48}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.34$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(840=2^{3}\cdot 3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(643,·)$, $\chi_{840}(769,·)$, $\chi_{840}(139,·)$, $\chi_{840}(659,·)$, $\chi_{840}(281,·)$, $\chi_{840}(673,·)$, $\chi_{840}(547,·)$, $\chi_{840}(41,·)$, $\chi_{840}(43,·)$, $\chi_{840}(433,·)$, $\chi_{840}(307,·)$, $\chi_{840}(827,·)$, $\chi_{840}(449,·)$, $\chi_{840}(83,·)$, $\chi_{840}(323,·)$, $\chi_{840}(713,·)$, $\chi_{840}(587,·)$, $\chi_{840}(811,·)$, $\chi_{840}(337,·)$, $\chi_{840}(419,·)$, $\chi_{840}(601,·)$, $\chi_{840}(97,·)$, $\chi_{840}(379,·)$, $\chi_{840}(209,·)$, $\chi_{840}(617,·)$, $\chi_{840}(491,·)$, $\chi_{840}(113,·)$, $\chi_{840}(211,·)$, $\chi_{840}(169,·)$, $\chi_{840}(377,·)$, $\chi_{840}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{22} a^{18} - \frac{2}{11} a^{17} - \frac{1}{2} a^{16} - \frac{1}{11} a^{15} + \frac{4}{11} a^{14} - \frac{7}{22} a^{12} + \frac{3}{11} a^{11} - \frac{1}{2} a^{10} - \frac{4}{11} a^{9} - \frac{1}{2} a^{8} - \frac{2}{11} a^{7} - \frac{3}{11} a^{6} - \frac{3}{22} a^{4} - \frac{5}{11} a^{3} - \frac{1}{2} a^{2} + \frac{3}{11} a - \frac{1}{11}$, $\frac{1}{22} a^{19} - \frac{5}{22} a^{17} - \frac{1}{11} a^{16} + \frac{5}{11} a^{14} - \frac{7}{22} a^{13} - \frac{9}{22} a^{11} - \frac{4}{11} a^{10} + \frac{1}{22} a^{9} - \frac{2}{11} a^{8} - \frac{1}{11} a^{6} - \frac{3}{22} a^{5} - \frac{7}{22} a^{3} + \frac{3}{11} a^{2} - \frac{4}{11}$, $\frac{1}{44} a^{20} - \frac{1}{44} a^{18} + \frac{1}{11} a^{17} - \frac{5}{11} a^{15} - \frac{19}{44} a^{14} - \frac{1}{2} a^{13} + \frac{7}{44} a^{12} - \frac{3}{22} a^{11} + \frac{1}{44} a^{10} - \frac{7}{22} a^{9} - \frac{9}{22} a^{7} - \frac{5}{44} a^{6} - \frac{19}{44} a^{4} + \frac{5}{22} a^{3} + \frac{4}{11} a - \frac{2}{11}$, $\frac{1}{44} a^{21} - \frac{1}{44} a^{19} + \frac{4}{11} a^{17} - \frac{5}{11} a^{16} - \frac{1}{4} a^{15} - \frac{5}{22} a^{14} + \frac{7}{44} a^{13} - \frac{1}{2} a^{12} + \frac{21}{44} a^{11} - \frac{7}{22} a^{10} - \frac{3}{11} a^{9} - \frac{9}{22} a^{8} + \frac{1}{4} a^{7} - \frac{5}{11} a^{6} - \frac{19}{44} a^{5} - \frac{1}{2} a^{4} - \frac{1}{11} a^{3} + \frac{4}{11} a^{2} + \frac{3}{11} a + \frac{2}{11}$, $\frac{1}{88} a^{22} - \frac{1}{88} a^{20} - \frac{1}{8} a^{16} - \frac{1}{4} a^{15} - \frac{3}{8} a^{14} - \frac{1}{4} a^{13} - \frac{43}{88} a^{12} + \frac{1}{4} a^{11} - \frac{3}{22} a^{10} - \frac{1}{4} a^{9} - \frac{3}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{4}{11} a^{2} + \frac{4}{11}$, $\frac{1}{88} a^{23} - \frac{1}{88} a^{21} - \frac{1}{8} a^{17} - \frac{1}{4} a^{16} - \frac{3}{8} a^{15} - \frac{1}{4} a^{14} - \frac{43}{88} a^{13} + \frac{1}{4} a^{12} - \frac{3}{22} a^{11} - \frac{1}{4} a^{10} - \frac{3}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{4}{11} a^{3} + \frac{4}{11} a$, $\frac{1}{176} a^{24} - \frac{1}{176} a^{22} - \frac{3}{176} a^{18} + \frac{17}{88} a^{17} + \frac{5}{16} a^{16} - \frac{19}{88} a^{15} + \frac{21}{176} a^{14} - \frac{3}{8} a^{13} + \frac{5}{44} a^{12} - \frac{31}{88} a^{11} + \frac{5}{16} a^{10} + \frac{3}{22} a^{9} - \frac{1}{16} a^{8} - \frac{27}{88} a^{7} - \frac{3}{11} a^{6} - \frac{1}{4} a^{5} + \frac{2}{11} a^{4} + \frac{1}{22} a^{3} + \frac{2}{11} a^{2} + \frac{3}{11} a - \frac{1}{11}$, $\frac{1}{176} a^{25} - \frac{1}{176} a^{23} - \frac{3}{176} a^{19} + \frac{1}{88} a^{18} + \frac{7}{176} a^{17} - \frac{19}{88} a^{16} + \frac{85}{176} a^{15} + \frac{15}{88} a^{14} + \frac{5}{44} a^{13} - \frac{7}{88} a^{12} + \frac{39}{176} a^{11} + \frac{3}{22} a^{10} + \frac{69}{176} a^{9} - \frac{27}{88} a^{8} + \frac{5}{11} a^{7} - \frac{7}{44} a^{6} + \frac{2}{11} a^{5} - \frac{9}{22} a^{4} + \frac{3}{11} a^{2} - \frac{2}{11} a + \frac{4}{11}$, $\frac{1}{10038688} a^{26} - \frac{4391}{2509672} a^{25} - \frac{21001}{10038688} a^{24} - \frac{3379}{627418} a^{23} + \frac{717}{627418} a^{22} - \frac{21645}{2509672} a^{21} + \frac{49}{11552} a^{20} + \frac{100235}{5019344} a^{19} - \frac{65961}{10038688} a^{18} - \frac{1980819}{5019344} a^{17} - \frac{243249}{912608} a^{16} + \frac{1680295}{5019344} a^{15} + \frac{872673}{2509672} a^{14} + \frac{918419}{5019344} a^{13} - \frac{19073}{10038688} a^{12} - \frac{308227}{2509672} a^{11} + \frac{599397}{10038688} a^{10} + \frac{746813}{5019344} a^{9} - \frac{515419}{1254836} a^{8} - \frac{10189}{1254836} a^{7} + \frac{1092}{313709} a^{6} + \frac{211139}{1254836} a^{5} - \frac{219113}{627418} a^{4} + \frac{34467}{627418} a^{3} - \frac{146929}{313709} a^{2} - \frac{80392}{313709} a - \frac{71352}{313709}$, $\frac{1}{10038688} a^{27} + \frac{3445}{10038688} a^{25} + \frac{81}{132088} a^{24} - \frac{1}{176} a^{23} - \frac{12647}{2509672} a^{22} - \frac{15459}{10038688} a^{21} - \frac{8627}{5019344} a^{20} + \frac{204493}{10038688} a^{19} - \frac{1141}{63536} a^{18} - \frac{4514561}{10038688} a^{17} + \frac{1114143}{5019344} a^{16} + \frac{25111}{456304} a^{15} + \frac{2072109}{5019344} a^{14} + \frac{4248583}{10038688} a^{13} + \frac{1128483}{2509672} a^{12} - \frac{1488193}{10038688} a^{11} + \frac{1869083}{5019344} a^{10} - \frac{1810333}{5019344} a^{9} - \frac{229265}{2509672} a^{8} + \frac{1497}{1254836} a^{7} - \frac{412837}{1254836} a^{6} + \frac{622727}{1254836} a^{5} - \frac{367405}{1254836} a^{4} + \frac{282097}{627418} a^{3} + \frac{2}{11} a^{2} - \frac{15193}{313709} a + \frac{97965}{313709}$, $\frac{1}{2145568786240} a^{28} - \frac{47423}{1072784393120} a^{27} + \frac{6647}{195051707840} a^{26} - \frac{733248693}{1072784393120} a^{25} - \frac{431672017}{1072784393120} a^{24} + \frac{1415325663}{268196098280} a^{23} - \frac{4738068503}{2145568786240} a^{22} - \frac{4536571549}{536392196560} a^{21} + \frac{2296341461}{429113757248} a^{20} - \frac{10586013893}{536392196560} a^{19} + \frac{3897929099}{429113757248} a^{18} + \frac{129515086797}{536392196560} a^{17} - \frac{144038372319}{1072784393120} a^{16} + \frac{24065099643}{214556878624} a^{15} + \frac{10989655915}{429113757248} a^{14} + \frac{222626964093}{1072784393120} a^{13} + \frac{877632991}{14797026112} a^{12} + \frac{142164445653}{536392196560} a^{11} - \frac{40760444473}{1072784393120} a^{10} + \frac{577877629}{1849628264} a^{9} - \frac{27928474489}{536392196560} a^{8} + \frac{49250578349}{268196098280} a^{7} - \frac{492951875}{1849628264} a^{6} + \frac{3941796687}{13409804914} a^{5} + \frac{28071563383}{134098049140} a^{4} - \frac{28213028053}{67049024570} a^{3} + \frac{628528287}{3047682935} a^{2} + \frac{2477532483}{33524512285} a + \frac{16221056974}{33524512285}$, $\frac{1}{103214427207876416380203200} a^{29} - \frac{1110104364491}{20642885441575283276040640} a^{28} - \frac{3130921833317313339}{103214427207876416380203200} a^{27} - \frac{4019849101151410749}{103214427207876416380203200} a^{26} - \frac{4559341668708973105885}{2064288544157528327604064} a^{25} + \frac{896805173982699726673}{2580360680196910409505080} a^{24} - \frac{240056136451718272112319}{103214427207876416380203200} a^{23} + \frac{213107668572844121908291}{103214427207876416380203200} a^{22} - \frac{1110609081497058302023561}{103214427207876416380203200} a^{21} + \frac{144977518806264713619153}{103214427207876416380203200} a^{20} + \frac{1868305438470032126992653}{103214427207876416380203200} a^{19} + \frac{587851499598998539120423}{103214427207876416380203200} a^{18} + \frac{232013569658778312583339}{469156487308529165364560} a^{17} + \frac{6537941382463532351678901}{51607213603938208190101600} a^{16} + \frac{2567870422087362368854089}{20642885441575283276040640} a^{15} - \frac{4826307824291939323562929}{103214427207876416380203200} a^{14} + \frac{21630183977273196186793191}{103214427207876416380203200} a^{13} + \frac{3817354557734425604349507}{103214427207876416380203200} a^{12} + \frac{12293486151505104579872669}{25803606801969104095050800} a^{11} + \frac{2085366610998130921505023}{12901803400984552047525400} a^{10} + \frac{3522536596857920936455673}{12901803400984552047525400} a^{9} + \frac{6834648497555915121788859}{25803606801969104095050800} a^{8} + \frac{615027423303677080806993}{1612725425123069005940675} a^{7} + \frac{177899159654220411281529}{645090170049227602376270} a^{6} - \frac{43330130510722168610701}{3225450850246138011881350} a^{5} + \frac{656163895384666390512171}{3225450850246138011881350} a^{4} + \frac{704004115690879967207888}{1612725425123069005940675} a^{3} + \frac{57115402150976225054}{16976057106558621115165} a^{2} - \frac{710817460200163747623023}{1612725425123069005940675} a + \frac{238596387095220048804359}{1612725425123069005940675}$, $\frac{1}{206428854415752832760406400} a^{30} - \frac{48780641851}{10864676548197517513705600} a^{28} - \frac{418651946037060691}{51607213603938208190101600} a^{27} - \frac{35759776820922519}{938312974617058330729120} a^{26} - \frac{2153832847707826881749}{2064288544157528327604064} a^{25} + \frac{56928409110458847944601}{206428854415752832760406400} a^{24} - \frac{391316255661508060704287}{103214427207876416380203200} a^{23} - \frac{181473122489969515073041}{206428854415752832760406400} a^{22} + \frac{1064656646218414695752039}{103214427207876416380203200} a^{21} + \frac{35525624779045496290363}{18766259492341166614582400} a^{20} - \frac{1039092023802949302976551}{103214427207876416380203200} a^{19} - \frac{19374234277705184912977}{5160721360393820819010160} a^{18} + \frac{252225016744575062166939}{3559118179581945392420800} a^{17} - \frac{2133337037295363996671401}{41285770883150566552081280} a^{16} - \frac{242225827825334818030701}{679042284262344844606600} a^{15} + \frac{75761337075146419922080821}{206428854415752832760406400} a^{14} - \frac{10685829241670440231328609}{103214427207876416380203200} a^{13} + \frac{13825646323212990601997209}{51607213603938208190101600} a^{12} - \frac{2001357098531100822973037}{25803606801969104095050800} a^{11} + \frac{21061102638633676508436471}{51607213603938208190101600} a^{10} - \frac{667951608093744636813573}{1612725425123069005940675} a^{9} - \frac{11672753784623496500084461}{25803606801969104095050800} a^{8} - \frac{126273678052561514837}{1719094390537581885080} a^{7} + \frac{3417460188574127280501673}{12901803400984552047525400} a^{6} + \frac{155484213495434190775054}{1612725425123069005940675} a^{5} - \frac{709119952460426450108141}{1612725425123069005940675} a^{4} + \frac{268840367076188478802703}{645090170049227602376270} a^{3} - \frac{60789067034017486864053}{293222804567830728352850} a^{2} + \frac{8214055025472858478627}{1612725425123069005940675} a + \frac{16211982173044259866470}{64509017004922760237627}$, $\frac{1}{206428854415752832760406400} a^{31} - \frac{1}{206428854415752832760406400} a^{29} - \frac{3886571450711}{51607213603938208190101600} a^{28} - \frac{1752089568381541523}{51607213603938208190101600} a^{27} - \frac{12361513303013553}{51607213603938208190101600} a^{26} - \frac{519552612201096502912159}{206428854415752832760406400} a^{25} + \frac{78871581935255680523373}{103214427207876416380203200} a^{24} + \frac{221116439917949675025807}{206428854415752832760406400} a^{23} - \frac{535626527688154480614757}{103214427207876416380203200} a^{22} - \frac{5447853326384976339297}{2172935309639503502741120} a^{21} - \frac{427483295890145559867199}{103214427207876416380203200} a^{20} + \frac{106848962010080832358}{13328309298537760379675} a^{19} - \frac{104868936701320684536023}{5432338274098758756852800} a^{18} - \frac{16437204743816483383783937}{41285770883150566552081280} a^{17} - \frac{4675805075388704054328681}{25803606801969104095050800} a^{16} - \frac{30609245006312822669569019}{206428854415752832760406400} a^{15} + \frac{3494410407172162098856551}{20642885441575283276040640} a^{14} + \frac{788935988705038562549671}{51607213603938208190101600} a^{13} - \frac{513268381592417176189073}{5160721360393820819010160} a^{12} + \frac{25625399857793075128364993}{51607213603938208190101600} a^{11} + \frac{12663880387383399227265351}{25803606801969104095050800} a^{10} + \frac{2030127408692564052276903}{25803606801969104095050800} a^{9} - \frac{155182908999212545029621}{889779544895486348105200} a^{8} - \frac{465647082056000576345967}{1612725425123069005940675} a^{7} + \frac{87641454598313592617183}{679042284262344844606600} a^{6} + \frac{308448799642403453699593}{6450901700492276023762700} a^{5} - \frac{429076966517203993350377}{6450901700492276023762700} a^{4} - \frac{360293762292293845349582}{1612725425123069005940675} a^{3} + \frac{507513427665262715924432}{1612725425123069005940675} a^{2} - \frac{645789471071773117260667}{1612725425123069005940675} a + \frac{218890775099314960271576}{1612725425123069005940675}$

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

Class group and class number

Not computed

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{14432947312558132091727}{4691564873085291653645600} a^{31} - \frac{19307346271358170092921}{2345782436542645826822800} a^{30} + \frac{139494822308011783945831}{4691564873085291653645600} a^{29} - \frac{50803868905985565896067}{469156487308529165364560} a^{28} + \frac{39954913720965059484231}{213252948776604166074800} a^{27} + \frac{44539963496654647744629}{293222804567830728352850} a^{26} - \frac{70238327210554363186707}{59386897127661919666400} a^{25} + \frac{1746680444103242578734191}{234578243654264582682280} a^{24} - \frac{25923774402203764655875833}{938312974617058330729120} a^{23} + \frac{91122198604939045833533973}{1172891218271322913411400} a^{22} - \frac{586770525084339713173901679}{4691564873085291653645600} a^{21} + \frac{241698080537910360827765793}{1172891218271322913411400} a^{20} + \frac{85041543571586112520863823}{2345782436542645826822800} a^{19} - \frac{5405533522880227905375842427}{2345782436542645826822800} a^{18} + \frac{71691515518362080064336171}{8514636793258242565600} a^{17} - \frac{1328988863626345784310034581}{93831297461705833072912} a^{16} + \frac{46019621238203788514934050673}{938312974617058330729120} a^{15} - \frac{77473816454016507091502712211}{1172891218271322913411400} a^{14} + \frac{16860145175685913630039798581}{2345782436542645826822800} a^{13} + \frac{6700146580655131879463744313}{586445609135661456705700} a^{12} - \frac{5784573597335708698417488687}{586445609135661456705700} a^{11} + \frac{46236412862086055387737779}{11728912182713229134114} a^{10} - \frac{11453110729489830740864107}{26656618597075520759350} a^{9} - \frac{31714172719880248539522927}{29322280456783072835285} a^{8} + \frac{225214149564969071349057219}{293222804567830728352850} a^{7} - \frac{2914283128240407776235786}{13328309298537760379675} a^{6} - \frac{330950399978600268757551}{7716389593890282325075} a^{5} + \frac{35056395949123325620556743}{586445609135661456705700} a^{4} - \frac{558360838402027954747308}{13328309298537760379675} a^{3} + \frac{1378754257745669516613648}{146611402283915364176425} a^{2} + \frac{10144769520475227253284}{2665661859707552075935} a - \frac{477947567707104796843968}{146611402283915364176425} \) (order $30$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3\times C_4$ (as 32T34):

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 $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{105}) \), \(\Q(\sqrt{-210}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-2}, \sqrt{105})\), \(\Q(\sqrt{6}, \sqrt{70})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{-3}, \sqrt{70})\), \(\Q(\sqrt{6}, \sqrt{-35})\), \(\Q(\sqrt{-2}, \sqrt{-35})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-10}, \sqrt{-42})\), \(\Q(\sqrt{14}, \sqrt{30})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-42})\), \(\Q(\sqrt{-10}, \sqrt{21})\), \(\Q(\sqrt{14}, \sqrt{-15})\), \(\Q(\sqrt{-7}, \sqrt{30})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{14})\), \(\Q(\sqrt{21}, \sqrt{30})\), \(\Q(\sqrt{-15}, \sqrt{-42})\), \(\Q(\sqrt{-7}, \sqrt{-10})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{14})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{30}, \sqrt{-35})\), \(\Q(\sqrt{-10}, \sqrt{14})\), 4.4.3528000.1, 4.4.8000.1, \(\Q(\zeta_{5})\), 4.0.55125.1, \(\Q(\zeta_{15})^+\), 4.4.6125.1, 4.0.392000.2, 4.0.72000.2, 8.0.497871360000.17, 8.0.497871360000.13, 8.0.497871360000.3, 8.8.497871360000.2, 8.0.497871360000.2, 8.0.121550625.1, 8.0.497871360000.12, 8.0.497871360000.20, 8.0.497871360000.18, 8.0.497871360000.15, 8.0.497871360000.4, 8.0.6146560000.1, 8.0.497871360000.8, 8.0.207360000.2, 8.0.796594176.1, 8.8.12446784000000.2, 8.0.3038765625.3, 8.8.3038765625.1, 8.0.12446784000000.1, 8.0.12446784000000.19, 8.0.12446784000000.12, 8.0.12446784000000.13, 8.0.12446784000000.2, 8.0.12446784000000.7, 8.0.64000000.1, 8.0.5184000000.2, 8.0.153664000000.3, 8.8.12446784000000.5, 8.8.153664000000.2, 8.0.153664000000.6, 8.0.12446784000000.4, 8.8.12446784000000.1, 8.8.5184000000.2, 8.0.5184000000.6, 8.0.12446784000000.9, 8.0.12446784000000.17, 8.0.5184000000.4, \(\Q(\zeta_{15})\), 8.0.3038765625.1, 8.0.12446784000000.10, 8.0.153664000000.4, 8.0.37515625.1, 8.0.3038765625.2, 16.0.247875891108249600000000.2, 16.0.154922431942656000000000000.19, 16.0.154922431942656000000000000.9, 16.16.154922431942656000000000000.2, 16.0.154922431942656000000000000.1, 16.0.154922431942656000000000000.15, 16.0.9234096523681640625.1, 16.0.154922431942656000000000000.6, 16.0.154922431942656000000000000.3, 16.0.154922431942656000000000000.10, 16.0.154922431942656000000000000.17, 16.0.154922431942656000000000000.12, 16.0.23612624896000000000000.1, 16.0.154922431942656000000000000.16, 16.0.26873856000000000000.1

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 R R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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
$2$2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 2 x^{6} + 8 x^{4} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$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}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$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}$