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Properties

Label 32.0.426...625.1
Degree $32$
Signature $[0, 16]$
Discriminant $4.266\times 10^{49}$
Root discriminant $35.56$
Ramified primes $3, 5, 7, 29$
Class number not computed
Class group not computed
Galois group $C_2^2\times C_2^2:C_4$ (as 32T262)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 - 14*x^30 + 32*x^29 + 110*x^28 - 256*x^27 - 822*x^26 + 1761*x^25 + 5076*x^24 - 6234*x^23 - 33617*x^22 + 7049*x^21 + 145847*x^20 + 80587*x^19 - 200297*x^18 - 1040496*x^17 + 909785*x^16 - 126545*x^15 + 4807173*x^14 - 6468687*x^13 + 4092091*x^12 - 12326083*x^11 + 15573178*x^10 - 9966741*x^9 + 15430857*x^8 - 15762104*x^7 + 7858456*x^6 - 7358464*x^5 + 5444944*x^4 - 823424*x^3 + 88384*x^2 - 5760*x + 256)
 
gp: K = bnfinit(x^32 - 2*x^31 - 14*x^30 + 32*x^29 + 110*x^28 - 256*x^27 - 822*x^26 + 1761*x^25 + 5076*x^24 - 6234*x^23 - 33617*x^22 + 7049*x^21 + 145847*x^20 + 80587*x^19 - 200297*x^18 - 1040496*x^17 + 909785*x^16 - 126545*x^15 + 4807173*x^14 - 6468687*x^13 + 4092091*x^12 - 12326083*x^11 + 15573178*x^10 - 9966741*x^9 + 15430857*x^8 - 15762104*x^7 + 7858456*x^6 - 7358464*x^5 + 5444944*x^4 - 823424*x^3 + 88384*x^2 - 5760*x + 256, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256, -5760, 88384, -823424, 5444944, -7358464, 7858456, -15762104, 15430857, -9966741, 15573178, -12326083, 4092091, -6468687, 4807173, -126545, 909785, -1040496, -200297, 80587, 145847, 7049, -33617, -6234, 5076, 1761, -822, -256, 110, 32, -14, -2, 1]);
 

\( x^{32} - 2 x^{31} - 14 x^{30} + 32 x^{29} + 110 x^{28} - 256 x^{27} - 822 x^{26} + 1761 x^{25} + 5076 x^{24} - 6234 x^{23} - 33617 x^{22} + 7049 x^{21} + 145847 x^{20} + 80587 x^{19} - 200297 x^{18} - 1040496 x^{17} + 909785 x^{16} - 126545 x^{15} + 4807173 x^{14} - 6468687 x^{13} + 4092091 x^{12} - 12326083 x^{11} + 15573178 x^{10} - 9966741 x^{9} + 15430857 x^{8} - 15762104 x^{7} + 7858456 x^{6} - 7358464 x^{5} + 5444944 x^{4} - 823424 x^{3} + 88384 x^{2} - 5760 x + 256 \)

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:  \(42655280577413385127292283461599409580230712890625\)\(\medspace = 3^{16}\cdot 5^{24}\cdot 7^{16}\cdot 29^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $35.56$
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:  $3, 5, 7, 29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $16$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{10} a^{20} + \frac{1}{5} a^{19} + \frac{1}{10} a^{17} + \frac{1}{5} a^{16} - \frac{1}{5} a^{15} + \frac{2}{5} a^{14} - \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{3}{10} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{21} + \frac{1}{10} a^{19} + \frac{1}{10} a^{18} - \frac{1}{10} a^{16} - \frac{1}{5} a^{15} - \frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{10} a^{6} + \frac{1}{5} a^{5} - \frac{1}{2} a^{4} + \frac{1}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{2} a - \frac{1}{5}$, $\frac{1}{20} a^{22} - \frac{1}{20} a^{20} + \frac{1}{10} a^{19} + \frac{1}{10} a^{17} - \frac{1}{20} a^{16} - \frac{3}{20} a^{15} - \frac{7}{20} a^{14} - \frac{7}{20} a^{13} + \frac{1}{20} a^{12} - \frac{3}{20} a^{11} + \frac{7}{20} a^{10} - \frac{1}{20} a^{9} - \frac{7}{20} a^{8} - \frac{3}{10} a^{7} + \frac{1}{20} a^{6} + \frac{3}{10} a^{5} - \frac{1}{20} a^{4} + \frac{9}{20} a^{3} + \frac{1}{4} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{20} a^{23} - \frac{1}{20} a^{21} - \frac{1}{5} a^{19} + \frac{1}{10} a^{18} - \frac{3}{20} a^{17} + \frac{3}{20} a^{16} - \frac{3}{20} a^{15} + \frac{1}{4} a^{14} + \frac{9}{20} a^{13} + \frac{1}{20} a^{12} - \frac{1}{4} a^{11} - \frac{9}{20} a^{10} + \frac{9}{20} a^{9} + \frac{3}{10} a^{8} + \frac{9}{20} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{3}{20} a^{4} + \frac{1}{20} a^{3} + \frac{1}{10} a^{2} + \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{20} a^{24} - \frac{1}{20} a^{20} + \frac{1}{10} a^{19} - \frac{3}{20} a^{18} - \frac{1}{20} a^{17} + \frac{1}{5} a^{16} + \frac{1}{5} a^{15} + \frac{2}{5} a^{14} + \frac{2}{5} a^{13} - \frac{1}{10} a^{12} + \frac{1}{10} a^{11} + \frac{1}{10} a^{10} + \frac{3}{20} a^{9} + \frac{2}{5} a^{8} - \frac{1}{10} a^{7} + \frac{2}{5} a^{6} + \frac{1}{20} a^{5} + \frac{1}{5} a^{4} + \frac{9}{20} a^{3} + \frac{7}{20} a^{2} + \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{40} a^{25} - \frac{1}{40} a^{24} - \frac{1}{40} a^{22} - \frac{1}{40} a^{21} + \frac{1}{8} a^{19} - \frac{1}{5} a^{18} - \frac{1}{40} a^{17} + \frac{3}{40} a^{16} - \frac{1}{8} a^{15} + \frac{11}{40} a^{14} - \frac{7}{40} a^{13} + \frac{11}{40} a^{12} + \frac{19}{40} a^{11} - \frac{1}{20} a^{10} - \frac{1}{20} a^{9} + \frac{1}{40} a^{8} + \frac{3}{10} a^{7} - \frac{11}{40} a^{5} + \frac{1}{20} a^{4} - \frac{9}{40} a^{3} - \frac{1}{4} a^{2} - \frac{1}{10} a + \frac{2}{5}$, $\frac{1}{440} a^{26} + \frac{1}{220} a^{25} - \frac{3}{440} a^{24} - \frac{3}{440} a^{23} - \frac{1}{44} a^{22} - \frac{21}{440} a^{21} - \frac{21}{440} a^{20} - \frac{1}{440} a^{19} + \frac{31}{440} a^{18} + \frac{1}{220} a^{17} - \frac{2}{11} a^{16} + \frac{1}{10} a^{15} + \frac{9}{44} a^{14} - \frac{109}{220} a^{13} - \frac{23}{110} a^{12} + \frac{131}{440} a^{11} - \frac{4}{11} a^{10} - \frac{41}{440} a^{9} - \frac{163}{440} a^{8} + \frac{71}{220} a^{7} - \frac{153}{440} a^{6} - \frac{21}{440} a^{5} - \frac{163}{440} a^{4} - \frac{137}{440} a^{3} - \frac{47}{110} a^{2} + \frac{13}{110} a + \frac{14}{55}$, $\frac{1}{440} a^{27} + \frac{1}{110} a^{25} - \frac{1}{55} a^{24} - \frac{1}{110} a^{23} + \frac{1}{44} a^{22} + \frac{1}{44} a^{21} + \frac{19}{440} a^{20} - \frac{1}{5} a^{19} + \frac{9}{55} a^{18} - \frac{51}{440} a^{17} - \frac{1}{88} a^{16} + \frac{101}{440} a^{15} - \frac{211}{440} a^{14} - \frac{107}{440} a^{13} - \frac{101}{220} a^{12} - \frac{59}{440} a^{11} + \frac{191}{440} a^{10} + \frac{19}{88} a^{9} + \frac{21}{88} a^{8} - \frac{217}{440} a^{7} + \frac{87}{440} a^{6} + \frac{1}{4} a^{5} + \frac{189}{440} a^{4} + \frac{37}{88} a^{3} + \frac{26}{55} a^{2} - \frac{53}{110} a + \frac{16}{55}$, $\frac{1}{880} a^{28} + \frac{3}{440} a^{25} - \frac{7}{440} a^{24} - \frac{1}{40} a^{23} - \frac{1}{55} a^{22} - \frac{7}{880} a^{21} - \frac{1}{220} a^{20} - \frac{83}{440} a^{19} + \frac{89}{880} a^{18} + \frac{37}{176} a^{17} - \frac{217}{880} a^{16} + \frac{119}{880} a^{15} + \frac{391}{880} a^{14} - \frac{47}{220} a^{13} - \frac{329}{880} a^{12} - \frac{91}{880} a^{11} + \frac{15}{176} a^{10} + \frac{45}{176} a^{9} + \frac{149}{880} a^{8} - \frac{17}{176} a^{7} + \frac{97}{440} a^{6} + \frac{59}{176} a^{5} - \frac{87}{880} a^{4} - \frac{161}{440} a^{3} - \frac{41}{220} a^{2} + \frac{23}{110} a + \frac{1}{11}$, $\frac{1}{1760} a^{29} - \frac{1}{880} a^{27} + \frac{1}{176} a^{25} + \frac{3}{440} a^{24} - \frac{17}{880} a^{23} + \frac{3}{160} a^{22} + \frac{7}{880} a^{21} - \frac{39}{880} a^{20} + \frac{51}{1760} a^{19} - \frac{73}{352} a^{18} + \frac{93}{1760} a^{17} - \frac{63}{352} a^{16} - \frac{119}{1760} a^{15} - \frac{35}{176} a^{14} - \frac{347}{1760} a^{13} - \frac{135}{352} a^{12} + \frac{771}{1760} a^{11} - \frac{23}{160} a^{10} + \frac{173}{352} a^{9} - \frac{145}{352} a^{8} - \frac{111}{440} a^{7} + \frac{731}{1760} a^{6} - \frac{137}{1760} a^{5} - \frac{81}{880} a^{4} + \frac{9}{55} a^{3} - \frac{31}{220} a^{2} + \frac{23}{110} a + \frac{4}{55}$, $\frac{1}{788329307332495206080} a^{30} - \frac{273647326495943}{9613772040640185440} a^{29} + \frac{2039916433878847}{35833150333295236640} a^{28} - \frac{76118053460467967}{98541163416561900760} a^{27} - \frac{188096218806253209}{394164653666247603040} a^{26} - \frac{1004532165772086261}{98541163416561900760} a^{25} + \frac{786067483935272425}{78832930733249520608} a^{24} + \frac{2255536336218377941}{157665861466499041216} a^{23} + \frac{2230243329975676297}{98541163416561900760} a^{22} + \frac{12468895811385734187}{394164653666247603040} a^{21} - \frac{14883716100705914329}{788329307332495206080} a^{20} - \frac{168852905661253897}{1747958552843670080} a^{19} + \frac{158686810906565794859}{788329307332495206080} a^{18} - \frac{21787505725541715561}{788329307332495206080} a^{17} - \frac{169817880117851536397}{788329307332495206080} a^{16} + \frac{17957623019501153303}{197082326833123801520} a^{15} + \frac{148188667037518430921}{788329307332495206080} a^{14} + \frac{31496076079829491367}{157665861466499041216} a^{13} + \frac{178555436550903161521}{788329307332495206080} a^{12} + \frac{1798125515017844659}{14333260133318094656} a^{11} - \frac{5864877093093421449}{19227544081280370880} a^{10} - \frac{309177294374850626007}{788329307332495206080} a^{9} - \frac{48212931678796372929}{394164653666247603040} a^{8} - \frac{167162946317890604973}{788329307332495206080} a^{7} - \frac{183105746058106056059}{788329307332495206080} a^{6} + \frac{88070177429046161951}{197082326833123801520} a^{5} + \frac{36528014430284199709}{98541163416561900760} a^{4} - \frac{11690413197754584429}{24635290854140475190} a^{3} - \frac{19881883107465415079}{49270581708280950380} a^{2} - \frac{429344957526095494}{12317645427070237595} a + \frac{5279795271823257642}{12317645427070237595}$, $\frac{1}{288455737035887715314436770652170257098514870217994059753795696802268410869242240} a^{31} - \frac{20889925525748514290042805533672996475748342894658607470761}{72113934258971928828609192663042564274628717554498514938448924200567102717310560} a^{30} + \frac{3088672774147775049511954201368211068485113729953924884460604901368307223161}{13111624410722168877928944120553193504477948646272457261536168036466745948601920} a^{29} - \frac{17756273659347998463249243334761481153531952727910409177013513589579148912851}{72113934258971928828609192663042564274628717554498514938448924200567102717310560} a^{28} + \frac{131845665850486877869498045571811334317793481460109352165014016496678927151783}{144227868517943857657218385326085128549257435108997029876897848401134205434621120} a^{27} - \frac{49724992952038930985432971698292585659140495332266680755322355765239526674167}{72113934258971928828609192663042564274628717554498514938448924200567102717310560} a^{26} - \frac{144126195050597204058445332843057108627859263238036317760230965847501410161209}{13111624410722168877928944120553193504477948646272457261536168036466745948601920} a^{25} + \frac{5960314693012913034673771651892887841414898855475336984639833216182842830517309}{288455737035887715314436770652170257098514870217994059753795696802268410869242240} a^{24} - \frac{207756391561954845273550372424884045352066416721536405919013638544379298804051}{28845573703588771531443677065217025709851487021799405975379569680226841086924224} a^{23} + \frac{60984093617327819022202825678590855000205147842912088416762208028501708314727}{144227868517943857657218385326085128549257435108997029876897848401134205434621120} a^{22} + \frac{1024186446197686093572325241212738706385005765168811980776040808513382748320543}{57691147407177543062887354130434051419702974043598811950759139360453682173848448} a^{21} + \frac{9831042507618616027385144016903158327843023448559645241574140768170282938769579}{288455737035887715314436770652170257098514870217994059753795696802268410869242240} a^{20} + \frac{36941243170342065365282645942015057089059538944322032325915769937612449078621413}{288455737035887715314436770652170257098514870217994059753795696802268410869242240} a^{19} - \frac{24395308491564165579193667494171621620518431077619764138314649152808164940615}{1407101156272623001533837905620342717553731074234117364652661935620821516435328} a^{18} - \frac{1267264392801332914042714759181298858546115496662351935390614126876815032467231}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{17} - \frac{17251460453225419532445921090521100684514074184649428115996885548218958739901299}{144227868517943857657218385326085128549257435108997029876897848401134205434621120} a^{16} + \frac{10725817611827609261664585017885278079043844423286202792399306676999346585646289}{288455737035887715314436770652170257098514870217994059753795696802268410869242240} a^{15} + \frac{22864808203041327921020023035694648115748819663818046594345878219221148571246753}{57691147407177543062887354130434051419702974043598811950759139360453682173848448} a^{14} - \frac{60625644037309444498960802201814666103985504464686599571527749466015652084493001}{288455737035887715314436770652170257098514870217994059753795696802268410869242240} a^{13} - \frac{36646045520553523588319418058038254443420380187596401135895827646678438346617441}{288455737035887715314436770652170257098514870217994059753795696802268410869242240} a^{12} + \frac{10281603340531812198380572742680961135125059348478133975241500885002534974358849}{288455737035887715314436770652170257098514870217994059753795696802268410869242240} a^{11} - \frac{83029105081459595800697279191414515222659507651418277405637446053070992174566561}{288455737035887715314436770652170257098514870217994059753795696802268410869242240} a^{10} - \frac{11699787871173739989526990738939861953508225249184242724002166841686699741289403}{36056967129485964414304596331521282137314358777249257469224462100283551358655280} a^{9} - \frac{125745480692280213823497352011627512242641632106295126112583206277039550953223297}{288455737035887715314436770652170257098514870217994059753795696802268410869242240} a^{8} - \frac{133551343301505639906055217392227209293363396158164686846228377816980331760800453}{288455737035887715314436770652170257098514870217994059753795696802268410869242240} a^{7} + \frac{3399062661061449389750201068041464829979883463543931269165695810597979114163019}{144227868517943857657218385326085128549257435108997029876897848401134205434621120} a^{6} - \frac{3034959994759567152566312187662941858361597202084590017553308950094371276544307}{9014241782371491103576149082880320534328589694312314367306115525070887839663820} a^{5} + \frac{1068440590703473895857464023748608889677815118482354737478075804007806673189049}{7211393425897192882860919266304256427462871755449851493844892420056710271731056} a^{4} - \frac{4131298301815072323612228344225588362907184802166146299461174998687034468904889}{9014241782371491103576149082880320534328589694312314367306115525070887839663820} a^{3} - \frac{461678264904253273056718599379533103954605283210223099397210015209516688448469}{9014241782371491103576149082880320534328589694312314367306115525070887839663820} a^{2} + \frac{377281144831000144994964273914202687611090811711442589308049308712410194860453}{2253560445592872775894037270720080133582147423578078591826528881267721959915955} a - \frac{902499289029543860064023555750542287594322284588310982843126081405894084785838}{2253560445592872775894037270720080133582147423578078591826528881267721959915955}$

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

Class group and class number

not computed

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{720893405312274704384024174360685296659707118301748692365836756778748803403}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{31} - \frac{596956557805948854765838033802918985478218753799444563787170535496758268653}{3517752890681557503834594764050856793884327685585293411631654839052053791088320} a^{30} - \frac{479777494337456926140951821115732645969997403064569164714690200752224402581}{319795717334687045803144978550077890353120698689572128330150439913823071917120} a^{29} + \frac{305150277902964982080450321436218825486080763764749327742502795180861052357}{109929777833798671994831086376589274808885240174540419113489213720376680971510} a^{28} + \frac{43385588945075487318662641549257719771732045915112142673300468661568652973001}{3517752890681557503834594764050856793884327685585293411631654839052053791088320} a^{27} - \frac{19524629652676093189439017191830204283900404707391320893013975636023288967357}{879438222670389375958648691012714198471081921396323352907913709763013447772080} a^{26} - \frac{65246452705993631676282428431600423903200728090584543857598436082288534426313}{703550578136311500766918952810171358776865537117058682326330967810410758217664} a^{25} + \frac{1057139610785117946265267629172011015936283835299492623258945253428503014210483}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{24} + \frac{50864052877728424274788011946392042436278705133253989834288543491402135351539}{87943822267038937595864869101271419847108192139632335290791370976301344777208} a^{23} - \frac{1588430148432129872380769971762083610120436105276001596479569457892433194012949}{3517752890681557503834594764050856793884327685585293411631654839052053791088320} a^{22} - \frac{25615415967767905633175080349919486927733946751837126482121395904752933010247107}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{21} - \frac{314522513511228076262093155603260373462475004380845358419675455367518728886851}{639591434669374091606289957100155780706241397379144256660300879827646143834240} a^{20} + \frac{105788061077183621515698657400877550192121863710169569808035404523677414585759173}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{19} + \frac{94497221338830408499709142479297854904052311270567200971282970091036122034311649}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{18} - \frac{119638420733514272990780685304610939167120321777014247366540493557393139051714163}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{17} - \frac{12454516549102420493126370701976242816154395605859983065069364169058296729982577}{109929777833798671994831086376589274808885240174540419113489213720376680971510} a^{16} + \frac{391591350375108354020000744113171819761768878137808464379681261148673707522516559}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{15} + \frac{19858911698605301473625583309729039502770814761642971884844387402279711111161965}{1407101156272623001533837905620342717553731074234117364652661935620821516435328} a^{14} + \frac{3464285953602823352464692217845569091951062010793820128039052909172814724256622223}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{13} - \frac{3478847387988679038819046464148892171584385035618038410001312558998836606079581837}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{12} + \frac{1514986832121584754699326432454806256899966699572218127257995052463790606989706697}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{11} - \frac{1618229660153217740034487933706175522895336443355222345676590555665802954492587573}{1407101156272623001533837905620342717553731074234117364652661935620821516435328} a^{10} + \frac{4160484814643741413561690604518680499742073474136097236885459683823111637378269411}{3517752890681557503834594764050856793884327685585293411631654839052053791088320} a^{9} - \frac{3770202377147965049975509288459902997888459012679425031916245422253191925650469003}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{8} + \frac{9188616291050342071131025261612071937268100375632243546483498479230230698864414847}{7035505781363115007669189528101713587768655371170586823263309678104107582176640} a^{7} - \frac{246576089343840664193652258959629536810002565021862986515535103086230369833052803}{219859555667597343989662172753178549617770480349080838226978427440753361943020} a^{6} + \frac{580191987481316088170877496167722494917407277619169600183393129350419750496597883}{1758876445340778751917297382025428396942163842792646705815827419526026895544160} a^{5} - \frac{243457380637994224799065036560752278636221878898445284265584889120018070569974627}{439719111335194687979324345506357099235540960698161676453956854881506723886040} a^{4} + \frac{147988898343866193410477705552382166395709041241326553875929475261891313868573703}{439719111335194687979324345506357099235540960698161676453956854881506723886040} a^{3} + \frac{696277462201536578645228476898001704747474893039124675576575551495364075758351}{9993616166708970181348280579689934073535021834049129010317201247306970997410} a^{2} + \frac{581495348273575027564300947129498206945572193231756198136694240971551868263301}{109929777833798671994831086376589274808885240174540419113489213720376680971510} a - \frac{18798842408387479295842271097217523103224547946640320697506112845384717720273}{54964888916899335997415543188294637404442620087270209556744606860188340485755} \) (order $30$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_2^2\times C_2^2:C_4$ (as 32T262):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 64
The 40 conjugacy class representatives for $C_2^2\times C_2^2:C_4$
Character table for $C_2^2\times C_2^2:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{105}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{5})\), 4.4.32625.1, 4.0.1598625.1, 4.4.6125.1, 4.4.177625.1, 4.0.55125.1, \(\Q(\zeta_{15})^+\), 4.0.3625.1, \(\Q(\sqrt{-15}, \sqrt{21})\), 4.0.35525.3, \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), 4.4.319725.1, \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), 4.0.6525.1, \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), 4.4.725.1, 8.0.102224075625.7, 8.0.102224075625.13, 8.0.102224075625.3, 8.0.1262025625.3, 8.0.121550625.1, 8.8.102224075625.1, 8.0.42575625.1, 8.0.31550640625.3, 8.0.2555601890625.18, 8.0.2555601890625.12, 8.0.31550640625.6, 8.0.2555601890625.11, 8.0.37515625.1, 8.0.3038765625.2, 8.0.31550640625.5, 8.0.2555601890625.15, 8.8.2555601890625.1, 8.8.2555601890625.2, 8.0.2555601890625.10, 8.0.3038765625.3, 8.8.2555601890625.3, 8.8.3038765625.1, 8.0.2555601890625.6, 8.0.1064390625.3, 8.0.2555601890625.7, 8.0.2555601890625.5, 8.0.1064390625.1, \(\Q(\zeta_{15})\), 8.0.2555601890625.14, 8.0.3038765625.1, 8.0.1064390625.2, 8.8.1064390625.1, 8.0.2555601890625.13, 8.8.31550640625.1, 8.0.13140625.1, Deg 16, Deg 16, Deg 16, Deg 16, Deg 16, Deg 16, Deg 16, Deg 16, 16.0.995442923847900390625.3, 16.0.9234096523681640625.1, Deg 16, Deg 16, Deg 16, 16.0.1132927402587890625.1, Deg 16

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{8}$ R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$

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

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
$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}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$