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Properties

Label 42.42.776...312.1
Degree $42$
Signature $[42, 0]$
Discriminant $7.767\times 10^{86}$
Root discriminant $117.17$
Ramified primes $2, 3, 7$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 105*x^40 + 4977*x^38 - 140819*x^36 - 2*x^35 + 2650095*x^34 + 28*x^33 - 34991649*x^32 + 3416*x^31 + 333300632*x^30 - 147910*x^29 - 2319579309*x^28 + 2698850*x^27 + 11820066804*x^26 - 26976474*x^25 - 43812582639*x^24 + 153194076*x^23 + 116368453593*x^22 - 445815310*x^21 - 216225058245*x^20 + 272316800*x^19 + 271556250546*x^18 + 1986860386*x^17 - 220004406804*x^16 - 5445985720*x^15 + 108239023559*x^14 + 5365080098*x^13 - 29954210552*x^12 - 2176838734*x^11 + 4303639312*x^10 + 340937478*x^9 - 323375283*x^8 - 25471098*x^7 + 12576760*x^6 + 955878*x^5 - 232127*x^4 - 16898*x^3 + 1505*x^2 + 112*x + 1)
 
gp: K = bnfinit(x^42 - 105*x^40 + 4977*x^38 - 140819*x^36 - 2*x^35 + 2650095*x^34 + 28*x^33 - 34991649*x^32 + 3416*x^31 + 333300632*x^30 - 147910*x^29 - 2319579309*x^28 + 2698850*x^27 + 11820066804*x^26 - 26976474*x^25 - 43812582639*x^24 + 153194076*x^23 + 116368453593*x^22 - 445815310*x^21 - 216225058245*x^20 + 272316800*x^19 + 271556250546*x^18 + 1986860386*x^17 - 220004406804*x^16 - 5445985720*x^15 + 108239023559*x^14 + 5365080098*x^13 - 29954210552*x^12 - 2176838734*x^11 + 4303639312*x^10 + 340937478*x^9 - 323375283*x^8 - 25471098*x^7 + 12576760*x^6 + 955878*x^5 - 232127*x^4 - 16898*x^3 + 1505*x^2 + 112*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 112, 1505, -16898, -232127, 955878, 12576760, -25471098, -323375283, 340937478, 4303639312, -2176838734, -29954210552, 5365080098, 108239023559, -5445985720, -220004406804, 1986860386, 271556250546, 272316800, -216225058245, -445815310, 116368453593, 153194076, -43812582639, -26976474, 11820066804, 2698850, -2319579309, -147910, 333300632, 3416, -34991649, 28, 2650095, -2, -140819, 0, 4977, 0, -105, 0, 1]);
 

\(x^{42} - 105 x^{40} + 4977 x^{38} - 140819 x^{36} - 2 x^{35} + 2650095 x^{34} + 28 x^{33} - 34991649 x^{32} + 3416 x^{31} + 333300632 x^{30} - 147910 x^{29} - 2319579309 x^{28} + 2698850 x^{27} + 11820066804 x^{26} - 26976474 x^{25} - 43812582639 x^{24} + 153194076 x^{23} + 116368453593 x^{22} - 445815310 x^{21} - 216225058245 x^{20} + 272316800 x^{19} + 271556250546 x^{18} + 1986860386 x^{17} - 220004406804 x^{16} - 5445985720 x^{15} + 108239023559 x^{14} + 5365080098 x^{13} - 29954210552 x^{12} - 2176838734 x^{11} + 4303639312 x^{10} + 340937478 x^{9} - 323375283 x^{8} - 25471098 x^{7} + 12576760 x^{6} + 955878 x^{5} - 232127 x^{4} - 16898 x^{3} + 1505 x^{2} + 112 x + 1\)  Toggle raw display

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[42, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(776\!\cdots\!312\)\(\medspace = 2^{42}\cdot 3^{21}\cdot 7^{76}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $117.17$
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:  $2, 3, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $42$
This field is Galois and abelian over $\Q$.
Conductor:  \(588=2^{2}\cdot 3\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{588}(1,·)$, $\chi_{588}(515,·)$, $\chi_{588}(263,·)$, $\chi_{588}(11,·)$, $\chi_{588}(527,·)$, $\chi_{588}(529,·)$, $\chi_{588}(275,·)$, $\chi_{588}(277,·)$, $\chi_{588}(23,·)$, $\chi_{588}(25,·)$, $\chi_{588}(155,·)$, $\chi_{588}(541,·)$, $\chi_{588}(289,·)$, $\chi_{588}(421,·)$, $\chi_{588}(169,·)$, $\chi_{588}(95,·)$, $\chi_{588}(431,·)$, $\chi_{588}(179,·)$, $\chi_{588}(443,·)$, $\chi_{588}(445,·)$, $\chi_{588}(575,·)$, $\chi_{588}(193,·)$, $\chi_{588}(323,·)$, $\chi_{588}(407,·)$, $\chi_{588}(71,·)$, $\chi_{588}(457,·)$, $\chi_{588}(205,·)$, $\chi_{588}(337,·)$, $\chi_{588}(85,·)$, $\chi_{588}(121,·)$, $\chi_{588}(347,·)$, $\chi_{588}(37,·)$, $\chi_{588}(107,·)$, $\chi_{588}(359,·)$, $\chi_{588}(361,·)$, $\chi_{588}(491,·)$, $\chi_{588}(109,·)$, $\chi_{588}(239,·)$, $\chi_{588}(373,·)$, $\chi_{588}(505,·)$, $\chi_{588}(191,·)$, $\chi_{588}(253,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{97} a^{37} + \frac{45}{97} a^{36} + \frac{16}{97} a^{35} + \frac{34}{97} a^{34} + \frac{18}{97} a^{33} - \frac{26}{97} a^{32} + \frac{45}{97} a^{31} - \frac{27}{97} a^{30} - \frac{24}{97} a^{29} + \frac{22}{97} a^{28} - \frac{6}{97} a^{27} - \frac{19}{97} a^{26} + \frac{20}{97} a^{25} + \frac{35}{97} a^{24} - \frac{37}{97} a^{23} + \frac{39}{97} a^{22} + \frac{46}{97} a^{21} - \frac{20}{97} a^{20} + \frac{12}{97} a^{19} - \frac{27}{97} a^{18} + \frac{39}{97} a^{17} + \frac{2}{97} a^{16} + \frac{18}{97} a^{15} - \frac{37}{97} a^{14} + \frac{33}{97} a^{13} - \frac{6}{97} a^{12} + \frac{22}{97} a^{11} + \frac{12}{97} a^{10} - \frac{34}{97} a^{9} - \frac{31}{97} a^{8} - \frac{46}{97} a^{7} + \frac{28}{97} a^{6} - \frac{24}{97} a^{5} - \frac{18}{97} a^{4} - \frac{48}{97} a^{3} + \frac{28}{97} a^{2} + \frac{11}{97} a + \frac{1}{97}$, $\frac{1}{97} a^{38} + \frac{28}{97} a^{36} - \frac{7}{97} a^{35} + \frac{40}{97} a^{34} + \frac{37}{97} a^{33} - \frac{46}{97} a^{32} - \frac{15}{97} a^{31} + \frac{27}{97} a^{30} + \frac{35}{97} a^{29} - \frac{26}{97} a^{28} - \frac{40}{97} a^{27} + \frac{2}{97} a^{26} + \frac{8}{97} a^{25} + \frac{37}{97} a^{24} - \frac{42}{97} a^{23} + \frac{37}{97} a^{22} + \frac{44}{97} a^{21} + \frac{39}{97} a^{20} + \frac{15}{97} a^{19} - \frac{7}{97} a^{18} - \frac{7}{97} a^{17} + \frac{25}{97} a^{16} + \frac{26}{97} a^{15} - \frac{48}{97} a^{14} - \frac{36}{97} a^{13} + \frac{1}{97} a^{12} - \frac{8}{97} a^{11} + \frac{8}{97} a^{10} + \frac{44}{97} a^{9} - \frac{9}{97} a^{8} - \frac{36}{97} a^{7} - \frac{23}{97} a^{6} - \frac{5}{97} a^{5} - \frac{14}{97} a^{4} - \frac{43}{97} a^{3} + \frac{12}{97} a^{2} - \frac{9}{97} a - \frac{45}{97}$, $\frac{1}{97} a^{39} - \frac{6}{97} a^{36} - \frac{20}{97} a^{35} - \frac{42}{97} a^{34} + \frac{32}{97} a^{33} + \frac{34}{97} a^{32} + \frac{28}{97} a^{31} + \frac{15}{97} a^{30} - \frac{33}{97} a^{29} + \frac{23}{97} a^{28} - \frac{24}{97} a^{27} - \frac{42}{97} a^{26} - \frac{38}{97} a^{25} + \frac{45}{97} a^{24} + \frac{6}{97} a^{23} + \frac{19}{97} a^{22} + \frac{12}{97} a^{21} - \frac{7}{97} a^{20} + \frac{45}{97} a^{19} - \frac{27}{97} a^{18} - \frac{30}{97} a^{16} + \frac{30}{97} a^{15} + \frac{30}{97} a^{14} + \frac{47}{97} a^{13} - \frac{34}{97} a^{12} - \frac{26}{97} a^{11} - \frac{1}{97} a^{10} - \frac{27}{97} a^{9} - \frac{41}{97} a^{8} + \frac{4}{97} a^{7} - \frac{13}{97} a^{6} - \frac{21}{97} a^{5} - \frac{24}{97} a^{4} - \frac{2}{97} a^{3} - \frac{17}{97} a^{2} + \frac{35}{97} a - \frac{28}{97}$, $\frac{1}{97} a^{40} - \frac{41}{97} a^{36} - \frac{43}{97} a^{35} + \frac{42}{97} a^{34} + \frac{45}{97} a^{33} - \frac{31}{97} a^{32} - \frac{6}{97} a^{31} - \frac{1}{97} a^{30} - \frac{24}{97} a^{29} + \frac{11}{97} a^{28} + \frac{19}{97} a^{27} + \frac{42}{97} a^{26} - \frac{29}{97} a^{25} + \frac{22}{97} a^{24} - \frac{9}{97} a^{23} - \frac{45}{97} a^{22} - \frac{22}{97} a^{21} + \frac{22}{97} a^{20} + \frac{45}{97} a^{19} + \frac{32}{97} a^{18} + \frac{10}{97} a^{17} + \frac{42}{97} a^{16} + \frac{41}{97} a^{15} + \frac{19}{97} a^{14} - \frac{30}{97} a^{13} + \frac{35}{97} a^{12} + \frac{34}{97} a^{11} + \frac{45}{97} a^{10} + \frac{46}{97} a^{9} + \frac{12}{97} a^{8} + \frac{2}{97} a^{7} - \frac{47}{97} a^{6} + \frac{26}{97} a^{5} - \frac{13}{97} a^{4} - \frac{14}{97} a^{3} + \frac{9}{97} a^{2} + \frac{38}{97} a + \frac{6}{97}$, $\frac{1}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{41} + \frac{1071747384332595811060972956072762205574584408202437822253854549042165755101400567690797199744667469420193071374281777226995301339586807531150025002857046834259}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{40} - \frac{1096398917602982573174667878019687442600752918536303729405143543820200776693418497733005647763029869448377134441643365175886248927593238312354053614278442488541}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{39} + \frac{1092985310414210273124478226496311826564023621972306302436829513871497525586112905465946740726391242689987416840019422153407640557705558224636554813375996182329}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{38} - \frac{1022276412639072217574396745929071061302477854460291917082636103484726958394075365550524762556168694340892601594655440410766607663938974012642569971708951830165}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{37} + \frac{95711415895993553418795045063501443180157643637243464623889333584355497812667210042471973137256367632528148776305220706415939940937671273590251755350614659128803}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{36} + \frac{28301646667721857370951916424556187840416528932385505066027815318167654534559621037737896926169921941107422645384593558472387840400091991145937003606402937849067}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{35} + \frac{74481558507943759304781334910948815497107256239486594970918186972372183084301660136779994742876820447553888085729662614254014848740455089613977097161397947380298}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{34} + \frac{69163793911268873729622583047594516281537740861918768691934951447205374399476808762986320845198288581165759110130706736573782193102410056364276913621402262536390}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{33} + \frac{57899832040853277889517807357940719899049047278156945701589717856634363835685669663286824727789248014645164643226796077580622399010577194502794149967149699634914}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{32} - \frac{59915040830993273157887168781667315908994698237397970529365910874161034570046903347266583662498988001269244038463823395607878966854657705163557607127739822482805}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{31} - \frac{51008057009162951064832745978260320939222547902187464761164335263078728922606958028241322271503944514492869001718226354983816671746576951618569392827151487135136}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{30} + \frac{20551394343166344977876660094809698422334089973341001182383468779777574198964522154356101914216925667125476485737189372800981851526350317350063616328148719284362}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{29} - \frac{42708119604647340361949945764997414321932087362542319159517930304700615451752478090694399455196089126175456621658095229449003260896714740222199216733793945609819}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{28} - \frac{69297932647633506168437708198228505566822475338750859880670975375311704925991113598879499791217156244909768967059766858300751092309418925274027598477674114172929}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{27} - \frac{88873865180674488569729563754693751353454154094563533909611896583851478785064111409846277674281341000649128665299489151338152586798516825377180596895984189402622}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{26} - \frac{30947029196276955586261109662731444832276606543609216103962719888479175952661715830609056504737172970678591922739078302927199896692122037993301207736557169177461}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{25} + \frac{86162350629587619077280518093106052679346191222273354896467605653684158021727153914501990454448361111702732572294665968007808552724753637390665173664266065857140}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{24} - \frac{488020330444174575985893729987349804356770097733630542747242385361018769612358060720164512784191603564700162749570594329972420826714282359728365343938175085925}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{23} + \frac{56997765054095970947971432070842149006077257449729691515499411411133389306777236408810826409641611281442290807076999017736511647855155894379987543648019064010442}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{22} - \frac{92933226154491325988089902520701347071825421992484811401515899420069737705640171248234353437314330573069860242762673172982999071584107454768146118817799088868913}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{21} - \frac{42482501649696833038046898004293131756861975203378054669854857634221885297653486640237486224928142858719045427416743626084389683790145228935648805853601334074138}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{20} - \frac{94013815665758357588538801863359567029335645136824072909621606477452086630865097805443392715808583746002721521451202458352762801510100491270855913770589227461775}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{19} - \frac{42226500026468377291785681164060988527549990967726912703314232771023610478328736897953972544275203742399452645175710323351395759831083248289359750957022395587120}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{18} + \frac{28763803921551616545064612037035812469477322882027395754567071721814504970643337790936003640998367708825182147153519671712868933939457861950163169220878699310376}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{17} - \frac{66777599395463695138667574197073049926433503658428921674731713181968555336166457236438072644752111603927635285818907752593757390653166076493252413900933557155998}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{16} - \frac{51713790193027243153641218467066104144052294880340081762940007911125638235089705864889542181349906778862835077897952563683978953487238244233137819954635422101079}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{15} + \frac{39800867092339304185640158032371704831549593023485166387117505418945660023701383730705805984018527440215662587407405608754261905494785438419668947251023653091193}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{14} + \frac{77162735236866646757919033225917727116643008474798687463777031393508707803429928198261006298293533895853533980461746733694450511367698608245379793641101305592998}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{13} + \frac{61220398946511473717516589481973089821089033241060753433193433020707550547760284579235845435223984792760050180305964508372507240493488151052828814277052773799178}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{12} - \frac{77046423971446996351618955141187448894385091582378878379694841235824890597231896262241789641124330774787888261251525366649393325211203702426880041416772563367421}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{11} + \frac{17106624157391620891215281043025645482782977480285364657988794660690290090297411770256487554781500610772184234409939376678520273652213728334921683228741644698388}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{10} - \frac{7686369531354931670396087029599754879941588247907778637523836642308984409275085720992223464709973983584731951547037810220791218395180502125411928996975678591270}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{9} - \frac{11229846426935021854600992939953772521996871141060348958032577679510855711051170154086650401867695953757452267150815794155924821386328583321965665009841248220810}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{8} + \frac{64818956558632612345636739877735382210002743213780723093826076760706309356693436621163727113894748519686992586440364972434787557014796156388032800512425858177415}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{7} - \frac{67870837164749267718982293715117015910917870207053472054265819793998380604058203614734458733877304196396791041264013320104210478971771367538100907647549439665213}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{6} + \frac{7515939903967980913622648272696912526641612030859369934166684596447244291081154993067918488184826288260792771486213486915933846128629871748349202417164168831976}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{5} + \frac{47744891773631696171304598202887509822498669435444162711050930129038668920199318058753730181464748697944732227994220425248428787145677657869407590326632598415395}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{4} + \frac{10607028863484042656111364512789596465782892586012233176902583122984407960386518582743563840081149756991434112791395153104587560625137074093640797950875377794573}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{3} - \frac{65228164177096506516901658525043659396306813144306160101007833325173079277547311563156280003046390388012959274278625183673542254127067474955014509979398850664568}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a^{2} + \frac{38395362660300032527791504203653798802250179172330222396861495419124913119427210123174464628223858988317852080471162039670788622031628174519121168136695746552055}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987} a - \frac{2844792063973981000536030770776620522771052304593326374432191856883999894051175869932448292826454687518989423123329731153515642320113833915866121068146356228863}{218776833351493208574716950527654024047585634742940644926501383070343493712550849439018062088039620673844816561888234266991770441415574203256727717842541698706987}$  Toggle raw display

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:  $41$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed  Toggle raw display
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

$C_{42}$ (as 42T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{7})^+\), 6.6.4148928.1, 7.7.13841287201.1, 14.14.6864701899232030692065067008.1, \(\Q(\zeta_{49})^+\)

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 $42$ R $21^{2}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{6}$ $42$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ $21^{2}$ $42$ $21^{2}$

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
2Data not computed
3Data not computed
7Data not computed