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Properties

Label 40.0.64587971698...0096.1
Degree $40$
Signature $[0, 20]$
Discriminant $2^{40}\cdot 3^{20}\cdot 7^{20}\cdot 11^{32}$
Root discriminant $62.41$
Ramified primes $2, 3, 7, 11$
Class number $2525$ (GRH)
Class group $[5, 505]$ (GRH)
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1048576, 0, 11796480, 0, 71434240, 0, 491175936, 0, 2380038144, 0, 3898278912, 0, 3879307776, 0, 3244057728, 0, 2244362704, 0, 1123030092, 0, 433488557, 0, 142400001, 0, 39413938, 0, 8845635, 0, 1674040, 0, 275385, 0, 37422, 0, 4086, 0, 377, 0, 27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 + 27*x^38 + 377*x^36 + 4086*x^34 + 37422*x^32 + 275385*x^30 + 1674040*x^28 + 8845635*x^26 + 39413938*x^24 + 142400001*x^22 + 433488557*x^20 + 1123030092*x^18 + 2244362704*x^16 + 3244057728*x^14 + 3879307776*x^12 + 3898278912*x^10 + 2380038144*x^8 + 491175936*x^6 + 71434240*x^4 + 11796480*x^2 + 1048576)
 
gp: K = bnfinit(x^40 + 27*x^38 + 377*x^36 + 4086*x^34 + 37422*x^32 + 275385*x^30 + 1674040*x^28 + 8845635*x^26 + 39413938*x^24 + 142400001*x^22 + 433488557*x^20 + 1123030092*x^18 + 2244362704*x^16 + 3244057728*x^14 + 3879307776*x^12 + 3898278912*x^10 + 2380038144*x^8 + 491175936*x^6 + 71434240*x^4 + 11796480*x^2 + 1048576, 1)
 

Normalized defining polynomial

\( x^{40} + 27 x^{38} + 377 x^{36} + 4086 x^{34} + 37422 x^{32} + 275385 x^{30} + 1674040 x^{28} + 8845635 x^{26} + 39413938 x^{24} + 142400001 x^{22} + 433488557 x^{20} + 1123030092 x^{18} + 2244362704 x^{16} + 3244057728 x^{14} + 3879307776 x^{12} + 3898278912 x^{10} + 2380038144 x^{8} + 491175936 x^{6} + 71434240 x^{4} + 11796480 x^{2} + 1048576 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(645879716982910580473318299047937246179670894442916176714774438893060096=2^{40}\cdot 3^{20}\cdot 7^{20}\cdot 11^{32}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.41$
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:  $2, 3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(924=2^{2}\cdot 3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{924}(1,·)$, $\chi_{924}(643,·)$, $\chi_{924}(265,·)$, $\chi_{924}(911,·)$, $\chi_{924}(785,·)$, $\chi_{924}(533,·)$, $\chi_{924}(155,·)$, $\chi_{924}(797,·)$, $\chi_{924}(419,·)$, $\chi_{924}(421,·)$, $\chi_{924}(295,·)$, $\chi_{924}(169,·)$, $\chi_{924}(839,·)$, $\chi_{924}(685,·)$, $\chi_{924}(559,·)$, $\chi_{924}(433,·)$, $\chi_{924}(181,·)$, $\chi_{924}(713,·)$, $\chi_{924}(575,·)$, $\chi_{924}(449,·)$, $\chi_{924}(323,·)$, $\chi_{924}(71,·)$, $\chi_{924}(841,·)$, $\chi_{924}(587,·)$, $\chi_{924}(463,·)$, $\chi_{924}(727,·)$, $\chi_{924}(335,·)$, $\chi_{924}(223,·)$, $\chi_{924}(97,·)$, $\chi_{924}(251,·)$, $\chi_{924}(113,·)$, $\chi_{924}(617,·)$, $\chi_{924}(881,·)$, $\chi_{924}(883,·)$, $\chi_{924}(757,·)$, $\chi_{924}(631,·)$, $\chi_{924}(377,·)$, $\chi_{924}(379,·)$, $\chi_{924}(125,·)$, $\chi_{924}(895,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{22} - \frac{1}{4} a^{20} + \frac{1}{4} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{14} + \frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{23} - \frac{1}{8} a^{21} - \frac{3}{8} a^{19} + \frac{1}{4} a^{17} - \frac{1}{4} a^{15} + \frac{1}{8} a^{13} - \frac{1}{2} a^{11} + \frac{3}{8} a^{9} - \frac{1}{4} a^{7} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{24} - \frac{1}{16} a^{22} + \frac{5}{16} a^{20} - \frac{3}{8} a^{18} + \frac{3}{8} a^{16} + \frac{1}{16} a^{14} - \frac{1}{4} a^{12} + \frac{3}{16} a^{10} - \frac{1}{8} a^{8} - \frac{7}{16} a^{6} + \frac{1}{16} a^{4}$, $\frac{1}{32} a^{25} - \frac{1}{32} a^{23} + \frac{5}{32} a^{21} - \frac{3}{16} a^{19} + \frac{3}{16} a^{17} - \frac{15}{32} a^{15} - \frac{1}{8} a^{13} + \frac{3}{32} a^{11} - \frac{1}{16} a^{9} + \frac{9}{32} a^{7} - \frac{15}{32} a^{5}$, $\frac{1}{64} a^{26} - \frac{1}{64} a^{24} + \frac{5}{64} a^{22} - \frac{3}{32} a^{20} + \frac{3}{32} a^{18} - \frac{15}{64} a^{16} - \frac{1}{16} a^{14} - \frac{29}{64} a^{12} + \frac{15}{32} a^{10} + \frac{9}{64} a^{8} + \frac{17}{64} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{128} a^{27} - \frac{1}{128} a^{25} + \frac{5}{128} a^{23} - \frac{3}{64} a^{21} + \frac{3}{64} a^{19} - \frac{15}{128} a^{17} - \frac{1}{32} a^{15} + \frac{35}{128} a^{13} + \frac{15}{64} a^{11} + \frac{9}{128} a^{9} - \frac{47}{128} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{256} a^{28} - \frac{1}{256} a^{26} + \frac{5}{256} a^{24} - \frac{3}{128} a^{22} + \frac{3}{128} a^{20} + \frac{113}{256} a^{18} + \frac{31}{64} a^{16} + \frac{35}{256} a^{14} + \frac{15}{128} a^{12} + \frac{9}{256} a^{10} - \frac{47}{256} a^{8} + \frac{3}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{512} a^{29} - \frac{1}{512} a^{27} + \frac{5}{512} a^{25} - \frac{3}{256} a^{23} + \frac{3}{256} a^{21} - \frac{143}{512} a^{19} - \frac{33}{128} a^{17} - \frac{221}{512} a^{15} + \frac{15}{256} a^{13} + \frac{9}{512} a^{11} - \frac{47}{512} a^{9} - \frac{5}{16} a^{7} - \frac{1}{8} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{5120} a^{30} - \frac{1}{1024} a^{28} + \frac{5}{1024} a^{26} + \frac{43}{2560} a^{24} - \frac{53}{512} a^{22} - \frac{27}{1024} a^{20} + \frac{7}{128} a^{18} - \frac{345}{1024} a^{16} + \frac{149}{512} a^{14} - \frac{1599}{5120} a^{12} + \frac{361}{1024} a^{10} + \frac{47}{256} a^{8} - \frac{131}{320} a^{6} - \frac{1}{2} a^{4} - \frac{2}{5}$, $\frac{1}{10240} a^{31} - \frac{1}{2048} a^{29} + \frac{5}{2048} a^{27} + \frac{43}{5120} a^{25} - \frac{53}{1024} a^{23} - \frac{27}{2048} a^{21} + \frac{7}{256} a^{19} - \frac{345}{2048} a^{17} + \frac{149}{1024} a^{15} + \frac{3521}{10240} a^{13} - \frac{663}{2048} a^{11} + \frac{47}{512} a^{9} + \frac{189}{640} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{3}{10} a$, $\frac{1}{449188925440} a^{32} + \frac{7887823}{89837785088} a^{30} + \frac{28643829}{89837785088} a^{28} + \frac{607731443}{224594462720} a^{26} - \frac{1053406149}{44918892544} a^{24} - \frac{3889054395}{89837785088} a^{22} + \frac{2220664369}{11229723136} a^{20} - \frac{33262944473}{89837785088} a^{18} - \frac{7360506547}{44918892544} a^{16} + \frac{2677003}{6704312320} a^{14} + \frac{1134545711}{3905990656} a^{12} - \frac{4422465453}{22459446272} a^{10} - \frac{6002348391}{28074307840} a^{8} + \frac{14696699}{350928848} a^{6} + \frac{42424063}{175464424} a^{4} + \frac{61668233}{438661060} a^{2} + \frac{142160}{21933053}$, $\frac{1}{898377850880} a^{33} + \frac{7887823}{179675570176} a^{31} + \frac{28643829}{179675570176} a^{29} + \frac{607731443}{449188925440} a^{27} - \frac{1053406149}{89837785088} a^{25} - \frac{3889054395}{179675570176} a^{23} + \frac{2220664369}{22459446272} a^{21} - \frac{33262944473}{179675570176} a^{19} + \frac{37558385997}{89837785088} a^{17} - \frac{6701635317}{13408624640} a^{15} - \frac{2771444945}{7811981312} a^{13} + \frac{18036980819}{44918892544} a^{11} + \frac{22071959449}{56148615680} a^{9} + \frac{14696699}{701857696} a^{7} + \frac{42424063}{350928848} a^{5} - \frac{376992827}{877322120} a^{3} - \frac{21790893}{43866106} a$, $\frac{1}{195846371491840} a^{34} - \frac{13}{39169274298368} a^{32} + \frac{2391276293}{195846371491840} a^{30} - \frac{6501019267}{97923185745920} a^{28} + \frac{21409719687}{19584637149184} a^{26} - \frac{3044518434447}{195846371491840} a^{24} - \frac{33582693}{688001024} a^{22} - \frac{15163064861241}{39169274298368} a^{20} + \frac{5179433818403}{19584637149184} a^{18} - \frac{86437138883639}{195846371491840} a^{16} - \frac{12020939963747}{39169274298368} a^{14} - \frac{1924886473561}{6120199109120} a^{12} - \frac{415675069857}{1530049777280} a^{10} - \frac{23802315911}{306009955456} a^{8} - \frac{83165531793}{765024888640} a^{6} + \frac{7947678649}{95628111080} a^{4} + \frac{322225577}{9562811108} a^{2} - \frac{2859358641}{11953513885}$, $\frac{1}{391692742983680} a^{35} - \frac{13}{78338548596736} a^{33} + \frac{2391276293}{391692742983680} a^{31} - \frac{6501019267}{195846371491840} a^{29} + \frac{21409719687}{39169274298368} a^{27} - \frac{3044518434447}{391692742983680} a^{25} - \frac{33582693}{1376002048} a^{23} - \frac{15163064861241}{78338548596736} a^{21} - \frac{14405203330781}{39169274298368} a^{19} - \frac{86437138883639}{391692742983680} a^{17} + \frac{27148334334621}{78338548596736} a^{15} - \frac{1924886473561}{12240398218240} a^{13} - \frac{415675069857}{3060099554560} a^{11} + \frac{282207639545}{612019910912} a^{9} - \frac{83165531793}{1530049777280} a^{7} - \frac{87680432431}{191256222160} a^{5} + \frac{322225577}{19125622216} a^{3} - \frac{2859358641}{23907027770} a$, $\frac{1}{10726555081507803931279360} a^{36} + \frac{11886484143}{10726555081507803931279360} a^{34} - \frac{45280177997}{40785380538052486430720} a^{32} + \frac{92988645605695088877}{5363277540753901965639680} a^{30} + \frac{644561377284582832859}{5363277540753901965639680} a^{28} - \frac{28534228650662726206671}{10726555081507803931279360} a^{26} + \frac{42613811978010281565987}{2681638770376950982819840} a^{24} + \frac{92541965112314529310615}{2145311016301560786255872} a^{22} - \frac{120424203399456773669045}{1072655508150780393127936} a^{20} - \frac{3481386773158114554474919}{10726555081507803931279360} a^{18} + \frac{9042826259249205651109}{27019030432009581690880} a^{16} - \frac{130575289237615011180383}{335204846297118872852480} a^{14} + \frac{131614356533381288641797}{670409692594237745704960} a^{12} - \frac{17488494061271294150621}{83801211574279718213120} a^{10} + \frac{1563525887857345378489}{10475151446784964776640} a^{8} - \frac{1372693289576643356617}{5237575723392482388320} a^{6} + \frac{391702688266177388283}{1309393930848120597080} a^{4} + \frac{3181763828025283327}{654696965424060298540} a^{2} + \frac{378467807351546843}{2442899124716642905}$, $\frac{1}{21453110163015607862558720} a^{37} + \frac{11886484143}{21453110163015607862558720} a^{35} - \frac{45280177997}{81570761076104972861440} a^{33} + \frac{92988645605695088877}{10726555081507803931279360} a^{31} + \frac{644561377284582832859}{10726555081507803931279360} a^{29} - \frac{28534228650662726206671}{21453110163015607862558720} a^{27} + \frac{42613811978010281565987}{5363277540753901965639680} a^{25} + \frac{92541965112314529310615}{4290622032603121572511744} a^{23} - \frac{120424203399456773669045}{2145311016301560786255872} a^{21} - \frac{3481386773158114554474919}{21453110163015607862558720} a^{19} + \frac{9042826259249205651109}{54038060864019163381760} a^{17} - \frac{130575289237615011180383}{670409692594237745704960} a^{15} + \frac{131614356533381288641797}{1340819385188475491409920} a^{13} - \frac{17488494061271294150621}{167602423148559436426240} a^{11} - \frac{8911625558927619398151}{20950302893569929553280} a^{9} + \frac{3864882433815839031703}{10475151446784964776640} a^{7} - \frac{917691242581943208797}{2618787861696241194160} a^{5} + \frac{3181763828025283327}{1309393930848120597080} a^{3} + \frac{378467807351546843}{4885798249433285810} a$, $\frac{1}{308381333169726221358123784166984903246479360} a^{38} - \frac{5409703319744280013}{308381333169726221358123784166984903246479360} a^{36} - \frac{327174599472520763903123194159}{308381333169726221358123784166984903246479360} a^{34} + \frac{143229938704615969924081207096447}{154190666584863110679061892083492451623239680} a^{32} - \frac{2082373443877611439486588982579398807813}{30838133316972622135812378416698490324647936} a^{30} + \frac{174596052560843875473549836138028155180201}{308381333169726221358123784166984903246479360} a^{28} - \frac{862783361415089099660694377255621466851}{287669154076237146789294574782635170938880} a^{26} + \frac{3935896217566598071358633841824789732970323}{308381333169726221358123784166984903246479360} a^{24} + \frac{2936203379011633465451906935960475348932409}{30838133316972622135812378416698490324647936} a^{22} - \frac{35076688576686270766303149343293884356270559}{308381333169726221358123784166984903246479360} a^{20} - \frac{36163993645855473711095075145347216457066843}{308381333169726221358123784166984903246479360} a^{18} + \frac{2606032766792958304754693069059988577765109}{77095333292431555339530946041746225811619840} a^{16} + \frac{1672202882867267948287218827206313727494453}{4818458330776972208720684127609139113226240} a^{14} + \frac{368860993596225146416080868317293141720767}{2409229165388486104360342063804569556613120} a^{12} + \frac{427088798132453643171221245966235169794759}{1204614582694243052180171031902284778306560} a^{10} + \frac{521411594890206305217913795854812276747}{2352762856824693461289396546684149957630} a^{8} - \frac{2349464454186412776352110455668982272049}{75288411418390190761260689493892798644160} a^{6} - \frac{211863566787099230340692592685359094319}{2352762856824693461289396546684149957630} a^{4} + \frac{1073357015951303061585272712902389995461}{4705525713649386922578793093368299915260} a^{2} + \frac{144982855952129300890447541293929992534}{1176381428412346730644698273342074978815}$, $\frac{1}{616762666339452442716247568333969806492958720} a^{39} - \frac{5409703319744280013}{616762666339452442716247568333969806492958720} a^{37} - \frac{327174599472520763903123194159}{616762666339452442716247568333969806492958720} a^{35} + \frac{143229938704615969924081207096447}{308381333169726221358123784166984903246479360} a^{33} - \frac{2082373443877611439486588982579398807813}{61676266633945244271624756833396980649295872} a^{31} + \frac{174596052560843875473549836138028155180201}{616762666339452442716247568333969806492958720} a^{29} - \frac{862783361415089099660694377255621466851}{575338308152474293578589149565270341877760} a^{27} + \frac{3935896217566598071358633841824789732970323}{616762666339452442716247568333969806492958720} a^{25} + \frac{2936203379011633465451906935960475348932409}{61676266633945244271624756833396980649295872} a^{23} - \frac{35076688576686270766303149343293884356270559}{616762666339452442716247568333969806492958720} a^{21} - \frac{36163993645855473711095075145347216457066843}{616762666339452442716247568333969806492958720} a^{19} - \frac{74489300525638597034776252972686237233854731}{154190666584863110679061892083492451623239680} a^{17} - \frac{3146255447909704260433465300402825385731787}{9636916661553944417441368255218278226452480} a^{15} - \frac{2040368171792260957944261195487276414892353}{4818458330776972208720684127609139113226240} a^{13} - \frac{777525784561789409008949785936049608511801}{2409229165388486104360342063804569556613120} a^{11} - \frac{1831351261934487156071482750829337680883}{4705525713649386922578793093368299915260} a^{9} - \frac{2349464454186412776352110455668982272049}{150576822836780381522521378987785597288320} a^{7} + \frac{2140899290037594230948703953998790863311}{4705525713649386922578793093368299915260} a^{5} + \frac{1073357015951303061585272712902389995461}{9411051427298773845157586186736599830520} a^{3} + \frac{72491427976064650445223770646964996267}{1176381428412346730644698273342074978815} a$

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

Class group and class number

$C_{5}\times C_{505}$, which has order $2525$ (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:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{623707563273105359562039726849512513}{935906929194920246913880983814825199534080} a^{39} - \frac{3344085493619318051347282927923330787}{187181385838984049382776196762965039906816} a^{37} - \frac{231916248763591676912233692882376219413}{935906929194920246913880983814825199534080} a^{35} - \frac{250361665715461365688415169306454818609}{93590692919492024691388098381482519953408} a^{33} - \frac{11427558337276160765646359398920942760371}{467953464597460123456940491907412599767040} a^{31} - \frac{167321717504192261904467196251296994495313}{935906929194920246913880983814825199534080} a^{29} - \frac{252881620244766139450056955842522809839243}{233976732298730061728470245953706299883520} a^{27} - \frac{123709197148775442562095590568755038105449}{21765277423137680160787929856158725570560} a^{25} - \frac{2354092536413742817452127577274631123541867}{93590692919492024691388098381482519953408} a^{23} - \frac{84187856018178467514350101065804453245557353}{935906929194920246913880983814825199534080} a^{21} - \frac{255000447344627861895823021978704034100747}{940609979090372107451136667150578089984} a^{19} - \frac{40625405234297123122433541926933943591863913}{58494183074682515432117561488426574970880} a^{17} - \frac{19841094479369489988927714680376946266441531}{14623545768670628858029390372106643742720} a^{15} - \frac{6899203954406263469861954805667593292715697}{3655886442167657214507347593026660935680} a^{13} - \frac{1003116816017173893892333866368367533931459}{456985805270957151813418449128332616960} a^{11} - \frac{488440996736658170154561494724506773995459}{228492902635478575906709224564166308480} a^{9} - \frac{129122051226392565898651058254296040619539}{114246451317739287953354612282083154240} a^{7} - \frac{237253200022849816045811928037134769389}{3570201603679352748542331633815098570} a^{5} - \frac{20947189623555005153421192022753032578}{1785100801839676374271165816907549285} a^{3} - \frac{12835500607881589270038415660134538651}{3570201603679352748542331633815098570} a \) (order $12$)
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:  \( 59944848702196220 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_{10}$ (as 40T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(i, \sqrt{21})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\zeta_{11})^+\), 8.0.49787136.1, 10.0.219503494144.1, 10.10.875463320250981.1, 10.0.896474439937004544.3, 10.10.53339349076992.1, 10.0.52089208083.1, 10.10.3689195226078208.1, 10.0.3602729712967.1, 20.0.803666421460365967723679147476647936.1, 20.0.2845086159957207322343768064.1, 20.0.13610161416118240236476132491264.1, 20.20.803666421460365967723679147476647936.1, 20.0.766436025104871719096831462361.1, 20.0.803666421460365967723679147476647936.5, 20.0.803666421460365967723679147476647936.7

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 ${\href{/LocalNumberField/5.10.0.1}{10} }^{4}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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

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
3Data not computed
7Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$