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Properties

Label 36.0.42776988625...5881.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{18}\cdot 7^{30}\cdot 19^{24}$
Root discriminant $62.42$
Ramified primes $3, 7, 19$
Class number Not computed
Class group Not computed
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13841287201, -11863960458, -1977326743, 9846280108, -6744817170, -1124136195, 5517032206, -341064451, -3562649419, 2549556730, 369199712, -2013093404, 1388580810, 231547784, -278192711, 36567144, 10161475, -15713252, 5259110, 3350415, -1441998, -19785, 97379, -95958, 30320, 6200, -334, 2020, -901, -492, 185, 12, 2, 5, -6, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 6*x^34 + 5*x^33 + 2*x^32 + 12*x^31 + 185*x^30 - 492*x^29 - 901*x^28 + 2020*x^27 - 334*x^26 + 6200*x^25 + 30320*x^24 - 95958*x^23 + 97379*x^22 - 19785*x^21 - 1441998*x^20 + 3350415*x^19 + 5259110*x^18 - 15713252*x^17 + 10161475*x^16 + 36567144*x^15 - 278192711*x^14 + 231547784*x^13 + 1388580810*x^12 - 2013093404*x^11 + 369199712*x^10 + 2549556730*x^9 - 3562649419*x^8 - 341064451*x^7 + 5517032206*x^6 - 1124136195*x^5 - 6744817170*x^4 + 9846280108*x^3 - 1977326743*x^2 - 11863960458*x + 13841287201)
 
gp: K = bnfinit(x^36 - x^35 - 6*x^34 + 5*x^33 + 2*x^32 + 12*x^31 + 185*x^30 - 492*x^29 - 901*x^28 + 2020*x^27 - 334*x^26 + 6200*x^25 + 30320*x^24 - 95958*x^23 + 97379*x^22 - 19785*x^21 - 1441998*x^20 + 3350415*x^19 + 5259110*x^18 - 15713252*x^17 + 10161475*x^16 + 36567144*x^15 - 278192711*x^14 + 231547784*x^13 + 1388580810*x^12 - 2013093404*x^11 + 369199712*x^10 + 2549556730*x^9 - 3562649419*x^8 - 341064451*x^7 + 5517032206*x^6 - 1124136195*x^5 - 6744817170*x^4 + 9846280108*x^3 - 1977326743*x^2 - 11863960458*x + 13841287201, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 6 x^{34} + 5 x^{33} + 2 x^{32} + 12 x^{31} + 185 x^{30} - 492 x^{29} - 901 x^{28} + 2020 x^{27} - 334 x^{26} + 6200 x^{25} + 30320 x^{24} - 95958 x^{23} + 97379 x^{22} - 19785 x^{21} - 1441998 x^{20} + 3350415 x^{19} + 5259110 x^{18} - 15713252 x^{17} + 10161475 x^{16} + 36567144 x^{15} - 278192711 x^{14} + 231547784 x^{13} + 1388580810 x^{12} - 2013093404 x^{11} + 369199712 x^{10} + 2549556730 x^{9} - 3562649419 x^{8} - 341064451 x^{7} + 5517032206 x^{6} - 1124136195 x^{5} - 6744817170 x^{4} + 9846280108 x^{3} - 1977326743 x^{2} - 11863960458 x + 13841287201 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(42776988625040032637276487792998794786679045219353738527006565881=3^{18}\cdot 7^{30}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.42$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(399=3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{399}(1,·)$, $\chi_{399}(134,·)$, $\chi_{399}(391,·)$, $\chi_{399}(11,·)$, $\chi_{399}(20,·)$, $\chi_{399}(277,·)$, $\chi_{399}(26,·)$, $\chi_{399}(286,·)$, $\chi_{399}(163,·)$, $\chi_{399}(292,·)$, $\chi_{399}(296,·)$, $\chi_{399}(172,·)$, $\chi_{399}(305,·)$, $\chi_{399}(178,·)$, $\chi_{399}(311,·)$, $\chi_{399}(58,·)$, $\chi_{399}(191,·)$, $\chi_{399}(64,·)$, $\chi_{399}(68,·)$, $\chi_{399}(197,·)$, $\chi_{399}(334,·)$, $\chi_{399}(83,·)$, $\chi_{399}(220,·)$, $\chi_{399}(349,·)$, $\chi_{399}(353,·)$, $\chi_{399}(229,·)$, $\chi_{399}(106,·)$, $\chi_{399}(235,·)$, $\chi_{399}(239,·)$, $\chi_{399}(368,·)$, $\chi_{399}(362,·)$, $\chi_{399}(115,·)$, $\chi_{399}(248,·)$, $\chi_{399}(121,·)$, $\chi_{399}(125,·)$, $\chi_{399}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4} a^{14} + \frac{1}{4} a^{7} + \frac{1}{4}$, $\frac{1}{4} a^{15} + \frac{1}{4} a^{8} + \frac{1}{4} a$, $\frac{1}{4} a^{16} + \frac{1}{4} a^{9} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} + \frac{1}{4} a^{10} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{18} + \frac{1}{4} a^{11} + \frac{1}{4} a^{4}$, $\frac{1}{28} a^{19} - \frac{1}{28} a^{16} + \frac{2}{7} a^{13} - \frac{1}{4} a^{12} - \frac{2}{7} a^{10} + \frac{1}{4} a^{9} + \frac{2}{7} a^{7} - \frac{1}{4} a^{5} - \frac{2}{7} a^{4} + \frac{1}{4} a^{2} + \frac{2}{7} a$, $\frac{1}{28} a^{20} - \frac{1}{28} a^{17} + \frac{1}{28} a^{14} - \frac{1}{4} a^{13} - \frac{2}{7} a^{11} + \frac{1}{4} a^{10} + \frac{2}{7} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{2}{7} a^{5} + \frac{1}{4} a^{3} + \frac{2}{7} a^{2} - \frac{1}{4}$, $\frac{1}{196} a^{21} + \frac{1}{196} a^{20} - \frac{2}{49} a^{18} + \frac{3}{98} a^{17} + \frac{1}{14} a^{16} + \frac{15}{196} a^{15} - \frac{13}{196} a^{14} + \frac{3}{28} a^{13} + \frac{19}{49} a^{12} + \frac{5}{49} a^{11} + \frac{3}{14} a^{10} + \frac{39}{98} a^{9} + \frac{71}{196} a^{8} + \frac{13}{28} a^{7} + \frac{13}{196} a^{6} - \frac{16}{49} a^{5} - \frac{1}{7} a^{4} - \frac{3}{98} a^{3} - \frac{45}{98} a^{2} + \frac{9}{28} a - \frac{1}{2}$, $\frac{1}{196} a^{22} - \frac{1}{196} a^{20} - \frac{1}{196} a^{19} + \frac{1}{14} a^{18} + \frac{2}{49} a^{17} - \frac{3}{98} a^{16} + \frac{3}{28} a^{15} - \frac{15}{196} a^{14} - \frac{85}{196} a^{13} + \frac{13}{28} a^{12} + \frac{11}{98} a^{11} - \frac{5}{49} a^{10} + \frac{3}{14} a^{9} + \frac{69}{196} a^{8} - \frac{71}{196} a^{7} - \frac{11}{28} a^{6} - \frac{13}{196} a^{5} - \frac{17}{98} a^{4} - \frac{3}{7} a^{3} + \frac{3}{98} a^{2} - \frac{2}{7} a + \frac{1}{4}$, $\frac{1}{196} a^{23} - \frac{97}{196} a^{2}$, $\frac{1}{196} a^{24} - \frac{97}{196} a^{3}$, $\frac{1}{196} a^{25} - \frac{97}{196} a^{4}$, $\frac{1}{30754948} a^{26} + \frac{11206}{7688737} a^{25} - \frac{17557}{15377474} a^{23} + \frac{44367}{30754948} a^{22} - \frac{186271}{30754948} a^{20} - \frac{270803}{30754948} a^{19} - \frac{131277}{2196782} a^{18} + \frac{742967}{30754948} a^{17} - \frac{537010}{7688737} a^{16} + \frac{525493}{4393564} a^{15} - \frac{1349233}{15377474} a^{14} + \frac{6513849}{30754948} a^{13} - \frac{1250621}{4393564} a^{12} + \frac{4994273}{15377474} a^{11} - \frac{705641}{30754948} a^{10} - \frac{233879}{1098391} a^{9} + \frac{7157135}{30754948} a^{8} - \frac{6874137}{15377474} a^{7} + \frac{1384491}{4393564} a^{6} - \frac{17}{49} a^{5} + \frac{1528865}{15377474} a^{4} + \frac{2122381}{4393564} a^{3} + \frac{23}{98} a^{2} - \frac{397842}{1098391} a - \frac{73301}{313826}$, $\frac{1}{215284636} a^{27} - \frac{1}{215284636} a^{26} - \frac{217787}{107642318} a^{25} - \frac{17557}{107642318} a^{24} + \frac{17557}{107642318} a^{23} - \frac{349239}{215284636} a^{22} + \frac{441381}{215284636} a^{21} - \frac{81946}{53821159} a^{20} - \frac{627779}{215284636} a^{19} + \frac{21245921}{215284636} a^{18} + \frac{23618863}{215284636} a^{17} - \frac{280869}{107642318} a^{16} + \frac{5852775}{107642318} a^{15} + \frac{8552423}{215284636} a^{14} + \frac{5921220}{53821159} a^{13} + \frac{20583637}{215284636} a^{12} + \frac{104974949}{215284636} a^{11} - \frac{44686273}{215284636} a^{10} - \frac{22552329}{107642318} a^{9} - \frac{38287915}{107642318} a^{8} + \frac{23489111}{215284636} a^{7} + \frac{45189135}{215284636} a^{6} - \frac{4747655}{107642318} a^{5} + \frac{12790483}{215284636} a^{4} - \frac{82250951}{215284636} a^{3} - \frac{3536102}{7688737} a^{2} + \frac{653245}{4393564} a - \frac{140339}{313826}$, $\frac{1}{6027969808} a^{28} - \frac{1}{753496226} a^{27} - \frac{3}{376748113} a^{26} + \frac{3224959}{1506992452} a^{25} - \frac{1028163}{1506992452} a^{24} + \frac{1519759}{1506992452} a^{23} + \frac{646987}{376748113} a^{22} - \frac{61139}{3013984904} a^{21} + \frac{10952099}{1506992452} a^{20} - \frac{22726801}{1506992452} a^{19} + \frac{29781277}{376748113} a^{18} + \frac{123595251}{1506992452} a^{17} + \frac{138053789}{1506992452} a^{16} - \frac{60364005}{753496226} a^{15} - \frac{164571713}{6027969808} a^{14} - \frac{278373129}{1506992452} a^{13} + \frac{352382335}{1506992452} a^{12} - \frac{157195214}{376748113} a^{11} - \frac{204688961}{1506992452} a^{10} + \frac{446915941}{1506992452} a^{9} + \frac{17688757}{753496226} a^{8} - \frac{1146949665}{3013984904} a^{7} - \frac{335112923}{1506992452} a^{6} + \frac{428690463}{1506992452} a^{5} + \frac{688402213}{1506992452} a^{4} + \frac{6284255}{15377474} a^{3} + \frac{6302113}{15377474} a^{2} + \frac{44353}{2196782} a - \frac{24699}{2510608}$, $\frac{1}{42195788656} a^{29} - \frac{1}{42195788656} a^{28} + \frac{23}{10548947164} a^{27} - \frac{109}{10548947164} a^{26} - \frac{10702445}{5274473582} a^{25} + \frac{3989753}{5274473582} a^{24} - \frac{15159047}{10548947164} a^{23} - \frac{15401387}{21097894328} a^{22} + \frac{3221667}{21097894328} a^{21} + \frac{56649079}{5274473582} a^{20} + \frac{23827301}{2637236791} a^{19} - \frac{895113447}{10548947164} a^{18} + \frac{183416033}{5274473582} a^{17} - \frac{243648728}{2637236791} a^{16} - \frac{687431421}{42195788656} a^{15} - \frac{5157722643}{42195788656} a^{14} + \frac{1137745513}{5274473582} a^{13} + \frac{433791588}{2637236791} a^{12} - \frac{1640281287}{10548947164} a^{11} + \frac{1908353739}{5274473582} a^{10} + \frac{988543046}{2637236791} a^{9} + \frac{396061273}{21097894328} a^{8} + \frac{8880035063}{21097894328} a^{7} + \frac{612605019}{10548947164} a^{6} - \frac{4334063207}{10548947164} a^{5} + \frac{551996181}{1506992452} a^{4} + \frac{10676695}{53821159} a^{3} - \frac{8826387}{30754948} a^{2} - \frac{6075063}{17574256} a + \frac{214849}{2510608}$, $\frac{1}{10928709261904} a^{30} + \frac{13}{1366088657738} a^{29} - \frac{3}{295370520592} a^{28} - \frac{3329}{2732177315476} a^{27} - \frac{20557}{1366088657738} a^{26} - \frac{30603017}{18460657537} a^{25} + \frac{1129221565}{1366088657738} a^{24} + \frac{717068631}{5464354630952} a^{23} - \frac{1139358727}{2732177315476} a^{22} - \frac{11788557555}{5464354630952} a^{21} - \frac{46093147817}{2732177315476} a^{20} + \frac{239572591}{683044328869} a^{19} + \frac{29838961309}{1366088657738} a^{18} - \frac{19896603287}{683044328869} a^{17} - \frac{1047010603805}{10928709261904} a^{16} + \frac{193617238725}{2732177315476} a^{15} - \frac{486499953273}{10928709261904} a^{14} + \frac{548131956823}{2732177315476} a^{13} + \frac{10438541958}{683044328869} a^{12} + \frac{553787958239}{1366088657738} a^{11} - \frac{46728291217}{683044328869} a^{10} + \frac{83543603513}{5464354630952} a^{9} + \frac{578775619543}{2732177315476} a^{8} - \frac{2323892815997}{5464354630952} a^{7} - \frac{332035366050}{683044328869} a^{6} - \frac{1329674555}{5274473582} a^{5} - \frac{12578936579}{27879360362} a^{4} - \frac{111732715}{3982765766} a^{3} + \frac{46383905}{123019792} a^{2} - \frac{813781}{40640467} a + \frac{9060941}{92892496}$, $\frac{1}{506361283376798382447096784} a^{31} + \frac{3289923860317}{253180641688399191223548392} a^{30} + \frac{54660705274771}{506361283376798382447096784} a^{29} - \frac{100719639319217}{506361283376798382447096784} a^{28} + \frac{146919795339977979}{126590320844199595611774196} a^{27} + \frac{441050214387013201}{63295160422099797805887098} a^{26} - \frac{92484415724475287229753}{126590320844199595611774196} a^{25} - \frac{301545384268471247616433}{253180641688399191223548392} a^{24} - \frac{196582550969139364796203}{126590320844199595611774196} a^{23} - \frac{152051801757900489074607}{253180641688399191223548392} a^{22} + \frac{352762169222985370221941}{253180641688399191223548392} a^{21} + \frac{454329197251788642949654}{31647580211049898902943549} a^{20} + \frac{502549323799876632026765}{63295160422099797805887098} a^{19} - \frac{113067047171417071680172}{31647580211049898902943549} a^{18} - \frac{17025100235690717371505757}{506361283376798382447096784} a^{17} + \frac{19049342372134138222737105}{253180641688399191223548392} a^{16} - \frac{27258790725592747164350883}{506361283376798382447096784} a^{15} - \frac{44130452210186965652389187}{506361283376798382447096784} a^{14} - \frac{15789685708459037170705296}{31647580211049898902943549} a^{13} + \frac{26411121412288693485232623}{63295160422099797805887098} a^{12} + \frac{10331683113271702390394194}{31647580211049898902943549} a^{11} - \frac{102841492826777196103919455}{253180641688399191223548392} a^{10} - \frac{21292946461061807141514407}{63295160422099797805887098} a^{9} - \frac{71949419637113186334699915}{253180641688399191223548392} a^{8} - \frac{57546085664563010546200025}{253180641688399191223548392} a^{7} - \frac{5608697602798500169650737}{18084331549171370801682028} a^{6} - \frac{313096101731420885974696}{645868983898977528631501} a^{5} + \frac{168073215162602258159215}{369067990799415730646572} a^{4} + \frac{72242094258275821204917}{210895994742523274655184} a^{3} - \frac{2806308033411040761919}{15063999624465948189656} a^{2} - \frac{1579877136210844152469}{4303999892704556625616} a - \frac{89341268832823295067}{614857127529222375088}$, $\frac{1}{3544528983637588677129677488} a^{32} - \frac{1}{3544528983637588677129677488} a^{31} - \frac{70942382596105}{3544528983637588677129677488} a^{30} - \frac{15433216937560669}{3544528983637588677129677488} a^{29} + \frac{15929813615733369}{3544528983637588677129677488} a^{28} + \frac{526095790260366079}{443066122954698584641209686} a^{27} + \frac{3077364223385479373}{443066122954698584641209686} a^{26} + \frac{4270011702289598345169921}{1772264491818794338564838744} a^{25} - \frac{3542364237028367269651243}{1772264491818794338564838744} a^{24} - \frac{1098036770329728865427587}{1772264491818794338564838744} a^{23} - \frac{2061379629675801684096063}{1772264491818794338564838744} a^{22} - \frac{849919998835340321837411}{1772264491818794338564838744} a^{21} - \frac{1753060004158613009521272}{221533061477349292320604843} a^{20} - \frac{1130292792920048143331842}{221533061477349292320604843} a^{19} + \frac{178037793450383044692219799}{3544528983637588677129677488} a^{18} + \frac{395712892880678828785602913}{3544528983637588677129677488} a^{17} + \frac{336218720554326417979975673}{3544528983637588677129677488} a^{16} + \frac{141292767267026669108962173}{3544528983637588677129677488} a^{15} + \frac{337522924365575005436060363}{3544528983637588677129677488} a^{14} + \frac{30791970920476701293354296}{221533061477349292320604843} a^{13} + \frac{38186346312389091828304350}{221533061477349292320604843} a^{12} + \frac{35581337657636993698242227}{1772264491818794338564838744} a^{11} + \frac{187707620800190149747586929}{1772264491818794338564838744} a^{10} - \frac{124729800031190904496704671}{1772264491818794338564838744} a^{9} - \frac{124712056886414884477292355}{1772264491818794338564838744} a^{8} + \frac{13161529981693027251108439}{253180641688399191223548392} a^{7} + \frac{1201370310295551068134275}{9042165774585685400841014} a^{6} + \frac{609249465503756893553533}{1291737967797955057263002} a^{5} + \frac{646309803588757432265221}{1476271963197662922586288} a^{4} + \frac{389338183527124674667}{814270249971132334576} a^{3} + \frac{7227203428944703277791}{30127999248931896379312} a^{2} - \frac{1209334702040615260707}{4303999892704556625616} a - \frac{22561524526671407247}{614857127529222375088}$, $\frac{1}{24811702885463120739907742416} a^{33} - \frac{1}{24811702885463120739907742416} a^{32} - \frac{3}{12405851442731560369953871208} a^{31} - \frac{513702524750291}{24811702885463120739907742416} a^{30} - \frac{42640102903980609}{12405851442731560369953871208} a^{29} + \frac{11109515435151663}{3101462860682890092488467802} a^{28} + \frac{7331216870974324285}{6202925721365780184976935604} a^{27} + \frac{158656149479679915867}{12405851442731560369953871208} a^{26} + \frac{17370180157108136989612997}{12405851442731560369953871208} a^{25} - \frac{13275483580307420085924995}{6202925721365780184976935604} a^{24} - \frac{23307864614805299477256041}{12405851442731560369953871208} a^{23} + \frac{14681850431574341586764301}{6202925721365780184976935604} a^{22} - \frac{10477538757587735608109511}{6202925721365780184976935604} a^{21} - \frac{5405632616305099010487740}{1550731430341445046244233901} a^{20} - \frac{239532863398274826565925493}{24811702885463120739907742416} a^{19} - \frac{2285372572945720812314532159}{24811702885463120739907742416} a^{18} - \frac{880772598630121955620455305}{12405851442731560369953871208} a^{17} - \frac{1064151134202033119437774917}{24811702885463120739907742416} a^{16} - \frac{762680820365075738146698743}{12405851442731560369953871208} a^{15} - \frac{51760490077755887355708425}{1550731430341445046244233901} a^{14} - \frac{180591330727958972778362811}{1550731430341445046244233901} a^{13} + \frac{5532088224882703015534582509}{12405851442731560369953871208} a^{12} + \frac{570715909949835254174340593}{12405851442731560369953871208} a^{11} - \frac{1830842285462017208775683767}{6202925721365780184976935604} a^{10} + \frac{4630801970749667643535274583}{12405851442731560369953871208} a^{9} - \frac{379032277124118069582172705}{886132245909397169282419372} a^{8} - \frac{2935515565646242768721079}{9042165774585685400841014} a^{7} - \frac{634356889620365826519587}{18084331549171370801682028} a^{6} + \frac{3132033155491001334487605}{10333903742383640458104016} a^{5} - \frac{18883771407253298223983}{1476271963197662922586288} a^{4} + \frac{5024211699471972051539}{15063999624465948189656} a^{3} + \frac{1595405562381544163693}{4303999892704556625616} a^{2} + \frac{95295708499089645399}{2151999946352278312808} a - \frac{36472861225527785791}{153714281882305593772}$, $\frac{1}{173681920198241845179354196912} a^{34} - \frac{1}{173681920198241845179354196912} a^{33} - \frac{3}{86840960099120922589677098456} a^{32} + \frac{5}{173681920198241845179354196912} a^{31} + \frac{6516128725339179}{173681920198241845179354196912} a^{30} + \frac{1051691061547930879}{173681920198241845179354196912} a^{29} - \frac{1097303962625304921}{173681920198241845179354196912} a^{28} + \frac{170119152977292372433}{86840960099120922589677098456} a^{27} + \frac{1241195504657823866579}{86840960099120922589677098456} a^{26} - \frac{22627913696062696073624738}{10855120012390115323709637307} a^{25} + \frac{220712255021073678891083553}{86840960099120922589677098456} a^{24} + \frac{90910823571329843423794423}{86840960099120922589677098456} a^{23} + \frac{176478909519110069987830853}{86840960099120922589677098456} a^{22} - \frac{111404344747515487379312133}{86840960099120922589677098456} a^{21} - \frac{918848164134417091570171489}{173681920198241845179354196912} a^{20} - \frac{224015886758033551177218967}{173681920198241845179354196912} a^{19} + \frac{10512053258793934634784515547}{86840960099120922589677098456} a^{18} + \frac{8177279899405113516542932071}{173681920198241845179354196912} a^{17} - \frac{7809784905127788783739650479}{173681920198241845179354196912} a^{16} - \frac{5006285961549153751805328471}{173681920198241845179354196912} a^{15} + \frac{13061789123557461047323672349}{173681920198241845179354196912} a^{14} - \frac{4951830339105098549676931321}{86840960099120922589677098456} a^{13} - \frac{14125793371203705820778298523}{86840960099120922589677098456} a^{12} + \frac{9338806124853598629164250747}{43420480049560461294838549228} a^{11} + \frac{18625233204246999577173020721}{86840960099120922589677098456} a^{10} - \frac{2615352709467866154557188827}{12405851442731560369953871208} a^{9} + \frac{39805003138298552736372851}{253180641688399191223548392} a^{8} - \frac{5005881857699613218440795}{253180641688399191223548392} a^{7} - \frac{10586346964224350421573011}{72337326196685483206728112} a^{6} + \frac{247899928280074343977027}{1476271963197662922586288} a^{5} - \frac{300387345635606344328485}{738135981598831461293144} a^{4} - \frac{8303408425769303982253}{210895994742523274655184} a^{3} + \frac{2265162153815331521795}{30127999248931896379312} a^{2} + \frac{964379428841940361721}{4303999892704556625616} a + \frac{74182606417303198935}{614857127529222375088}$, $\frac{1}{1215773441387692916255479378384} a^{35} - \frac{1}{1215773441387692916255479378384} a^{34} - \frac{3}{607886720693846458127739689192} a^{33} + \frac{5}{1215773441387692916255479378384} a^{32} + \frac{1}{607886720693846458127739689192} a^{31} + \frac{14203795105323753}{607886720693846458127739689192} a^{30} + \frac{1471439586368305041}{607886720693846458127739689192} a^{29} - \frac{3747761218935023991}{75985840086730807265967461149} a^{28} - \frac{1392252572493654720995}{607886720693846458127739689192} a^{27} + \frac{3265559131301270179629}{303943360346923229063869844596} a^{26} + \frac{1465535661821354890787981577}{607886720693846458127739689192} a^{25} + \frac{13916013401232548125554571}{8214685414781708893618103908} a^{24} - \frac{271884564415189293469285609}{303943360346923229063869844596} a^{23} - \frac{92864090780697419212823746}{75985840086730807265967461149} a^{22} - \frac{627384500960970833202790393}{1215773441387692916255479378384} a^{21} - \frac{15476050389417830083416188983}{1215773441387692916255479378384} a^{20} - \frac{168489133644303472029026721}{16429370829563417787236207816} a^{19} - \frac{39722175723320833113626546881}{1215773441387692916255479378384} a^{18} + \frac{7878537479637768030765839607}{607886720693846458127739689192} a^{17} + \frac{33789498620996872339619519}{16429370829563417787236207816} a^{16} + \frac{1786132475253610279133194823}{607886720693846458127739689192} a^{15} - \frac{8075267517923747403431792918}{75985840086730807265967461149} a^{14} - \frac{138474747917257962072354997851}{607886720693846458127739689192} a^{13} - \frac{93580426529393691638462368505}{303943360346923229063869844596} a^{12} - \frac{34251884163779113785049657515}{607886720693846458127739689192} a^{11} + \frac{18154856744531873278190710889}{43420480049560461294838549228} a^{10} - \frac{274263214593197283524638461}{886132245909397169282419372} a^{9} - \frac{364978874265428020790052275}{886132245909397169282419372} a^{8} - \frac{210873442219358124043342191}{506361283376798382447096784} a^{7} - \frac{32296355124718395272406455}{72337326196685483206728112} a^{6} + \frac{1171501436281070533501641}{5166951871191820229052008} a^{5} + \frac{458486467018634793675259}{1476271963197662922586288} a^{4} - \frac{5854010928172179026393}{15063999624465948189656} a^{3} - \frac{7446381073853422914309}{15063999624465948189656} a^{2} + \frac{762723953362369004607}{2151999946352278312808} a + \frac{142776243792877136315}{307428563764611187544}$

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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{8693241726592649}{1772264491818794338564838744} a^{35} - \frac{26079725179777947}{886132245909397169282419372} a^{34} + \frac{43466208632963245}{1772264491818794338564838744} a^{33} + \frac{8693241726592649}{886132245909397169282419372} a^{32} + \frac{26079725179777947}{443066122954698584641209686} a^{31} + \frac{43466208632963245}{47899040319426874015265912} a^{30} - \frac{1699340710971734725}{1772264491818794338564838744} a^{29} - \frac{7832610795659976749}{1772264491818794338564838744} a^{28} + \frac{4390087071929287745}{443066122954698584641209686} a^{27} - \frac{1451771368340972383}{886132245909397169282419372} a^{26} + \frac{6737262338109302975}{221533061477349292320604843} a^{25} + \frac{32947386143786139710}{221533061477349292320604843} a^{24} - \frac{417093044800188706371}{886132245909397169282419372} a^{23} - \frac{840616208679861928025}{886132245909397169282419372} a^{22} - \frac{171995787560635560465}{1772264491818794338564838744} a^{21} - \frac{6267818591631573336351}{886132245909397169282419372} a^{20} + \frac{29125967479401910099335}{1772264491818794338564838744} a^{19} + \frac{22859357248370333141195}{886132245909397169282419372} a^{18} - \frac{34149774486716348771137}{443066122954698584641209686} a^{17} + \frac{88336158473728037997275}{1772264491818794338564838744} a^{16} - \frac{205208327343171253944765}{1772264491818794338564838744} a^{15} - \frac{2418396483299129817981439}{1772264491818794338564838744} a^{14} + \frac{251612607196107718329977}{221533061477349292320604843} a^{13} + \frac{6035634319118909544232845}{886132245909397169282419372} a^{12} - \frac{625010984970758324885257}{63295160422099797805887098} a^{11} + \frac{1169658287829587737652}{645868983898977528631501} a^{10} + \frac{32308911005176542604195}{2583475935595910114526004} a^{9} - \frac{48725580799151704138296783}{221533061477349292320604843} a^{8} - \frac{25201707765392089451}{15063999624465948189656} a^{7} + \frac{203830438763417841103}{7531999812232974094828} a^{6} - \frac{1695182136685566555}{307428563764611187544} a^{5} - \frac{5085546410056699665}{153714281882305593772} a^{4} + \frac{3712014217255061123}{76857140941152796886} a^{3} - \frac{2981781912221278607}{307428563764611187544} a^{2} - \frac{19361510308988271515}{307428563764611187544} a + \frac{20872473385548950249}{307428563764611187544} \) (order $42$)
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

$C_6^2$ (as 36T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), 3.3.17689.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 3.3.361.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.0.8448319467.2, 6.0.8448319467.1, 6.0.64827.1, 6.0.3518667.1, 6.6.59138236269.1, 6.0.2190305047.1, 6.6.59138236269.2, 6.0.2190305047.2, \(\Q(\zeta_{21})^+\), \(\Q(\zeta_{7})\), 6.6.1206902781.1, 6.0.44700103.1, 9.9.5534900853769.1, 12.0.3497330989008067040361.1, 12.0.3497330989008067040361.2, \(\Q(\zeta_{21})\), 12.0.1456614322785533961.1, 18.0.602991213815902363206590020563.2, 18.18.206825986338854510579860377053109.1, 18.0.10507848719141112156676338823.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
19Data not computed