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Properties

Label 36.36.7061066244...1072.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{99}\cdot 3^{54}\cdot 7^{24}$
Root discriminant $127.91$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3457, -170472, -2193228, 12546916, 56713698, -247363308, -485300914, 1926063060, 1857934743, -7708877992, -3207637326, 17833884876, 1143439675, -25142360904, 4488876612, 22118582240, -7935268665, -12265171104, 6317132340, 4262622420, -2962642644, -877781932, 885664602, 79991748, -173279528, 6821832, 22076274, -2937888, -1758891, 379620, 78226, -25380, -1233, 872, -30, -12, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 12*x^35 - 30*x^34 + 872*x^33 - 1233*x^32 - 25380*x^31 + 78226*x^30 + 379620*x^29 - 1758891*x^28 - 2937888*x^27 + 22076274*x^26 + 6821832*x^25 - 173279528*x^24 + 79991748*x^23 + 885664602*x^22 - 877781932*x^21 - 2962642644*x^20 + 4262622420*x^19 + 6317132340*x^18 - 12265171104*x^17 - 7935268665*x^16 + 22118582240*x^15 + 4488876612*x^14 - 25142360904*x^13 + 1143439675*x^12 + 17833884876*x^11 - 3207637326*x^10 - 7708877992*x^9 + 1857934743*x^8 + 1926063060*x^7 - 485300914*x^6 - 247363308*x^5 + 56713698*x^4 + 12546916*x^3 - 2193228*x^2 - 170472*x + 3457)
 
gp: K = bnfinit(x^36 - 12*x^35 - 30*x^34 + 872*x^33 - 1233*x^32 - 25380*x^31 + 78226*x^30 + 379620*x^29 - 1758891*x^28 - 2937888*x^27 + 22076274*x^26 + 6821832*x^25 - 173279528*x^24 + 79991748*x^23 + 885664602*x^22 - 877781932*x^21 - 2962642644*x^20 + 4262622420*x^19 + 6317132340*x^18 - 12265171104*x^17 - 7935268665*x^16 + 22118582240*x^15 + 4488876612*x^14 - 25142360904*x^13 + 1143439675*x^12 + 17833884876*x^11 - 3207637326*x^10 - 7708877992*x^9 + 1857934743*x^8 + 1926063060*x^7 - 485300914*x^6 - 247363308*x^5 + 56713698*x^4 + 12546916*x^3 - 2193228*x^2 - 170472*x + 3457, 1)
 

Normalized defining polynomial

\( x^{36} - 12 x^{35} - 30 x^{34} + 872 x^{33} - 1233 x^{32} - 25380 x^{31} + 78226 x^{30} + 379620 x^{29} - 1758891 x^{28} - 2937888 x^{27} + 22076274 x^{26} + 6821832 x^{25} - 173279528 x^{24} + 79991748 x^{23} + 885664602 x^{22} - 877781932 x^{21} - 2962642644 x^{20} + 4262622420 x^{19} + 6317132340 x^{18} - 12265171104 x^{17} - 7935268665 x^{16} + 22118582240 x^{15} + 4488876612 x^{14} - 25142360904 x^{13} + 1143439675 x^{12} + 17833884876 x^{11} - 3207637326 x^{10} - 7708877992 x^{9} + 1857934743 x^{8} + 1926063060 x^{7} - 485300914 x^{6} - 247363308 x^{5} + 56713698 x^{4} + 12546916 x^{3} - 2193228 x^{2} - 170472 x + 3457 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7061066244864387270901673616087860193289967390430413386582332193634959491072=2^{99}\cdot 3^{54}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $127.91$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1008=2^{4}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1008}(611,·)$, $\chi_{1008}(1,·)$, $\chi_{1008}(515,·)$, $\chi_{1008}(961,·)$, $\chi_{1008}(779,·)$, $\chi_{1008}(529,·)$, $\chi_{1008}(659,·)$, $\chi_{1008}(25,·)$, $\chi_{1008}(793,·)$, $\chi_{1008}(155,·)$, $\chi_{1008}(947,·)$, $\chi_{1008}(673,·)$, $\chi_{1008}(169,·)$, $\chi_{1008}(683,·)$, $\chi_{1008}(179,·)$, $\chi_{1008}(841,·)$, $\chi_{1008}(697,·)$, $\chi_{1008}(827,·)$, $\chi_{1008}(193,·)$, $\chi_{1008}(11,·)$, $\chi_{1008}(289,·)$, $\chi_{1008}(457,·)$, $\chi_{1008}(337,·)$, $\chi_{1008}(851,·)$, $\chi_{1008}(121,·)$, $\chi_{1008}(347,·)$, $\chi_{1008}(107,·)$, $\chi_{1008}(865,·)$, $\chi_{1008}(995,·)$, $\chi_{1008}(361,·)$, $\chi_{1008}(323,·)$, $\chi_{1008}(491,·)$, $\chi_{1008}(625,·)$, $\chi_{1008}(275,·)$, $\chi_{1008}(505,·)$, $\chi_{1008}(443,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{6} a^{18} + \frac{1}{6} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{6} a^{6} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{19} + \frac{1}{6} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{20} + \frac{1}{6} a^{17} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{21} - \frac{1}{6} a^{15} - \frac{1}{2} a^{11} + \frac{1}{3} a^{9} - \frac{1}{2} a^{5} - \frac{1}{6} a^{3} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{22} - \frac{1}{6} a^{16} + \frac{1}{3} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{23} - \frac{1}{6} a^{17} + \frac{1}{3} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{24} - \frac{1}{4} a^{16} - \frac{1}{6} a^{15} + \frac{1}{6} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} + \frac{1}{3} a^{6} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{5}{12}$, $\frac{1}{12} a^{25} - \frac{1}{4} a^{17} - \frac{1}{6} a^{16} + \frac{1}{6} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{12} a^{26} - \frac{1}{12} a^{18} - \frac{1}{6} a^{17} + \frac{1}{6} a^{15} + \frac{1}{6} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{3} + \frac{1}{12} a^{2} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{12} a^{27} - \frac{1}{12} a^{19} + \frac{1}{6} a^{16} - \frac{1}{6} a^{15} + \frac{1}{3} a^{9} - \frac{1}{2} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{4} + \frac{5}{12} a^{3} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{12} a^{28} - \frac{1}{12} a^{20} + \frac{1}{6} a^{17} - \frac{1}{6} a^{16} + \frac{1}{3} a^{10} - \frac{1}{2} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{5} + \frac{5}{12} a^{4} - \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{12} a^{29} - \frac{1}{12} a^{21} - \frac{1}{6} a^{17} - \frac{1}{6} a^{15} - \frac{1}{6} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{5}{12} a^{5} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{204} a^{30} + \frac{7}{204} a^{29} - \frac{5}{204} a^{28} - \frac{7}{204} a^{27} - \frac{1}{68} a^{26} - \frac{1}{34} a^{25} - \frac{1}{102} a^{24} + \frac{1}{102} a^{23} + \frac{1}{204} a^{22} + \frac{11}{204} a^{21} + \frac{5}{204} a^{20} - \frac{1}{12} a^{19} + \frac{11}{204} a^{18} + \frac{11}{102} a^{17} + \frac{4}{51} a^{16} - \frac{4}{51} a^{15} + \frac{3}{17} a^{14} - \frac{1}{34} a^{13} + \frac{1}{34} a^{12} - \frac{35}{102} a^{11} + \frac{1}{17} a^{10} + \frac{2}{51} a^{9} + \frac{7}{34} a^{8} - \frac{31}{102} a^{7} + \frac{11}{68} a^{6} + \frac{19}{204} a^{5} + \frac{9}{68} a^{4} - \frac{71}{204} a^{3} - \frac{1}{204} a^{2} - \frac{10}{51} a + \frac{13}{51}$, $\frac{1}{204} a^{31} - \frac{1}{68} a^{29} - \frac{1}{34} a^{28} - \frac{5}{204} a^{27} - \frac{1}{102} a^{26} + \frac{1}{34} a^{25} - \frac{1}{204} a^{24} - \frac{13}{204} a^{23} + \frac{1}{51} a^{22} + \frac{13}{204} a^{21} + \frac{4}{51} a^{20} + \frac{11}{204} a^{19} - \frac{1}{51} a^{18} - \frac{3}{17} a^{17} - \frac{3}{68} a^{16} + \frac{23}{102} a^{15} + \frac{7}{102} a^{14} - \frac{5}{51} a^{13} - \frac{11}{51} a^{12} - \frac{2}{51} a^{11} - \frac{2}{51} a^{10} - \frac{4}{17} a^{9} - \frac{7}{17} a^{8} + \frac{25}{204} a^{7} + \frac{13}{102} a^{6} - \frac{89}{204} a^{5} - \frac{15}{34} a^{4} - \frac{65}{204} a^{3} + \frac{3}{34} a^{2} + \frac{5}{17} a - \frac{7}{204}$, $\frac{1}{204} a^{32} - \frac{1}{102} a^{29} - \frac{1}{68} a^{28} - \frac{1}{34} a^{27} - \frac{1}{68} a^{26} - \frac{1}{102} a^{25} - \frac{1}{102} a^{24} + \frac{5}{102} a^{23} + \frac{4}{51} a^{22} - \frac{1}{102} a^{21} + \frac{3}{68} a^{20} - \frac{1}{51} a^{19} - \frac{1}{68} a^{18} - \frac{7}{51} a^{17} - \frac{25}{204} a^{16} - \frac{7}{102} a^{14} - \frac{7}{51} a^{13} + \frac{11}{51} a^{12} - \frac{41}{102} a^{11} - \frac{23}{102} a^{10} - \frac{47}{102} a^{9} + \frac{83}{204} a^{8} + \frac{11}{51} a^{7} + \frac{5}{102} a^{6} + \frac{13}{51} a^{5} - \frac{35}{204} a^{4} - \frac{7}{34} a^{3} - \frac{15}{68} a^{2} - \frac{19}{51} a - \frac{33}{68}$, $\frac{1}{204} a^{33} - \frac{1}{34} a^{29} + \frac{1}{204} a^{28} - \frac{2}{51} a^{26} + \frac{1}{68} a^{25} + \frac{1}{34} a^{24} - \frac{7}{102} a^{23} + \frac{7}{102} a^{21} - \frac{11}{204} a^{20} + \frac{7}{102} a^{19} - \frac{1}{34} a^{18} - \frac{8}{51} a^{17} - \frac{3}{17} a^{16} - \frac{1}{17} a^{15} + \frac{11}{51} a^{14} - \frac{3}{17} a^{13} + \frac{8}{51} a^{12} + \frac{43}{102} a^{11} - \frac{1}{102} a^{10} - \frac{71}{204} a^{9} + \frac{5}{17} a^{8} - \frac{1}{17} a^{7} + \frac{25}{102} a^{6} + \frac{5}{51} a^{5} + \frac{21}{68} a^{4} + \frac{1}{3} a^{3} + \frac{29}{102} a^{2} - \frac{13}{102} a - \frac{25}{51}$, $\frac{1}{42690377963372816057469013761521861739939657683000749428} a^{34} - \frac{41825252696822069291876577082984902918107524929662645}{21345188981686408028734506880760930869969828841500374714} a^{33} + \frac{17098804665034307094355059400122089481304553871869929}{14230125987790938685823004587173953913313219227666916476} a^{32} - \frac{4766013585867134309652731047533064985578601386058835}{7115062993895469342911502293586976956656609613833458238} a^{31} - \frac{12545591863638622206535652751343329312181820344815389}{7115062993895469342911502293586976956656609613833458238} a^{30} - \frac{238937191907009918659653346436277030826133425326270884}{10672594490843204014367253440380465434984914420750187357} a^{29} + \frac{939872144901776508046899774438552623603665423875328987}{42690377963372816057469013761521861739939657683000749428} a^{28} + \frac{351037639353275610495008703504655595775430173078139825}{14230125987790938685823004587173953913313219227666916476} a^{27} + \frac{445888399260867972083320881925216214634209616510738003}{14230125987790938685823004587173953913313219227666916476} a^{26} + \frac{283920868082829127807790529509383586120797323954768169}{7115062993895469342911502293586976956656609613833458238} a^{25} - \frac{685344689440991699816614919023257361219127252607366979}{42690377963372816057469013761521861739939657683000749428} a^{24} - \frac{117416517361284553129773633523352547342288830855074854}{3557531496947734671455751146793488478328304806916729119} a^{23} - \frac{147263461580030609315381956702316806801374386235339568}{10672594490843204014367253440380465434984914420750187357} a^{22} - \frac{272978634559301565726401740042461804137535302915858771}{3557531496947734671455751146793488478328304806916729119} a^{21} + \frac{507972152726314022619487847338020129026186641072637843}{14230125987790938685823004587173953913313219227666916476} a^{20} + \frac{880464406316104405732402059338561000690281782318676939}{14230125987790938685823004587173953913313219227666916476} a^{19} + \frac{851252570895746300866523674935973683892390600579817171}{21345188981686408028734506880760930869969828841500374714} a^{18} - \frac{2311122539656092237431275673473189144444948409088082694}{10672594490843204014367253440380465434984914420750187357} a^{17} + \frac{3744973313149997251886955983683646306817087160489861073}{21345188981686408028734506880760930869969828841500374714} a^{16} - \frac{25995318373972611753167898757119822095104810472222297}{10672594490843204014367253440380465434984914420750187357} a^{15} - \frac{624679025144303012655016679859467556769018001479080901}{3557531496947734671455751146793488478328304806916729119} a^{14} - \frac{314879139536194879572738452446574931991662023874128065}{10672594490843204014367253440380465434984914420750187357} a^{13} - \frac{2936955184348445886153947237141233763540601551745452099}{21345188981686408028734506880760930869969828841500374714} a^{12} - \frac{6660477973266627748565125494232154299854086657349409015}{21345188981686408028734506880760930869969828841500374714} a^{11} - \frac{6792688625272201747245262832272349389433531189837640451}{42690377963372816057469013761521861739939657683000749428} a^{10} - \frac{602962754003986749441854090706373157063191611934289587}{7115062993895469342911502293586976956656609613833458238} a^{9} + \frac{9088880642053935530341174995019300570572657517027844921}{42690377963372816057469013761521861739939657683000749428} a^{8} - \frac{2877111880483384369324849405951915173125735219555360048}{10672594490843204014367253440380465434984914420750187357} a^{7} + \frac{6327712066204949007937847147507066328081129266983311373}{21345188981686408028734506880760930869969828841500374714} a^{6} - \frac{26236802353674558996321760851201105154652382453444265}{10672594490843204014367253440380465434984914420750187357} a^{5} + \frac{2670111528929425737720505333892424984079048537448809117}{42690377963372816057469013761521861739939657683000749428} a^{4} + \frac{4810552978972816086545517392302783563640531895397410105}{14230125987790938685823004587173953913313219227666916476} a^{3} + \frac{8075331096608996697169364673210932488647183605418644479}{21345188981686408028734506880760930869969828841500374714} a^{2} - \frac{319104598145840288138366372905937244727455245348866457}{10672594490843204014367253440380465434984914420750187357} a - \frac{732898798776047814368359908838088119320015543409233262}{10672594490843204014367253440380465434984914420750187357}$, $\frac{1}{1347953230001594106812794293020596424337064239545045593352184492359028} a^{35} - \frac{1171177500815}{112329435833466175567732857751716368694755353295420466112682041029919} a^{34} + \frac{1214357534846216433288323723855070497791958826676510477069557831703}{673976615000797053406397146510298212168532119772522796676092246179514} a^{33} + \frac{349238727203458049210388465519942125857874649596584432277250761762}{336988307500398526703198573255149106084266059886261398338046123089757} a^{32} - \frac{223922966019603141313608245227858388066262561697515770558647379827}{1347953230001594106812794293020596424337064239545045593352184492359028} a^{31} + \frac{2002478077038926806000409525979268630792553512336813378476195264499}{1347953230001594106812794293020596424337064239545045593352184492359028} a^{30} - \frac{13504570006567115033024856127907964293568257731405363467095058631093}{336988307500398526703198573255149106084266059886261398338046123089757} a^{29} - \frac{3613282637538216389467287991992200834822524935779975639981920094419}{449317743333864702270931431006865474779021413181681864450728164119676} a^{28} - \frac{1745679982439695605387690122504663681768633140490064728248704643374}{112329435833466175567732857751716368694755353295420466112682041029919} a^{27} + \frac{1740296864796979702225094068878647762932105233357361757915262989623}{673976615000797053406397146510298212168532119772522796676092246179514} a^{26} + \frac{13983614442245671167503891490233555346789409093188315358572607371119}{1347953230001594106812794293020596424337064239545045593352184492359028} a^{25} + \frac{50062584422390831003046672248153857319199435668284613187834117697195}{1347953230001594106812794293020596424337064239545045593352184492359028} a^{24} - \frac{39850338637373389883462065258759474115737425269794537899418090075489}{1347953230001594106812794293020596424337064239545045593352184492359028} a^{23} - \frac{25095121206193229331443979479401622168117048993753677459846770654635}{449317743333864702270931431006865474779021413181681864450728164119676} a^{22} - \frac{2713900044378174762794942137831166472149294425389925081238219298151}{39645683235341003141552773324135188951090124692501340980946602716442} a^{21} + \frac{10824328193963962098762259639780456819510809666749864510364352194053}{449317743333864702270931431006865474779021413181681864450728164119676} a^{20} + \frac{36550745097138941125083548438926314748351729761642370742653474625621}{1347953230001594106812794293020596424337064239545045593352184492359028} a^{19} - \frac{917899026172908902184084758333283732536632770182629305501997581221}{39645683235341003141552773324135188951090124692501340980946602716442} a^{18} - \frac{283036373459677210362911560830368288484650612247215220807173568484927}{1347953230001594106812794293020596424337064239545045593352184492359028} a^{17} - \frac{28629326948961914439367159581976823107591647741977434858016954492323}{1347953230001594106812794293020596424337064239545045593352184492359028} a^{16} + \frac{18299427764749104980359146972696678158530003987584733769214893824742}{112329435833466175567732857751716368694755353295420466112682041029919} a^{15} + \frac{90571945741592412728069777635280671980508640299410551041040273312251}{673976615000797053406397146510298212168532119772522796676092246179514} a^{14} - \frac{20619068050456830416520408173493788745625433103932935934170644634755}{224658871666932351135465715503432737389510706590840932225364082059838} a^{13} + \frac{100493559872915911358544288568222535573254251625196124228682044462107}{673976615000797053406397146510298212168532119772522796676092246179514} a^{12} - \frac{627271228710563106133409827111819510210946042125366278226617505539749}{1347953230001594106812794293020596424337064239545045593352184492359028} a^{11} + \frac{41925369590049957307174985752231672959940687712555183552588156171086}{336988307500398526703198573255149106084266059886261398338046123089757} a^{10} - \frac{21638416240211433642741470377287719397832803830245022359307280902982}{112329435833466175567732857751716368694755353295420466112682041029919} a^{9} + \frac{327093393827147828548104046610472220631142934304322886135779751307767}{673976615000797053406397146510298212168532119772522796676092246179514} a^{8} - \frac{123323289016409433553279268102041815660776991906677373859778846591935}{449317743333864702270931431006865474779021413181681864450728164119676} a^{7} - \frac{561514733018896677719564552358681661964809727084131320013431722777585}{1347953230001594106812794293020596424337064239545045593352184492359028} a^{6} + \frac{153935472799556365115875860870156133169035797364829998599762833043236}{336988307500398526703198573255149106084266059886261398338046123089757} a^{5} + \frac{27116865906820334577096377028866180940852441760055278312823260721329}{449317743333864702270931431006865474779021413181681864450728164119676} a^{4} - \frac{195332116685041086634228389635152140888172630909554490761118417367177}{449317743333864702270931431006865474779021413181681864450728164119676} a^{3} + \frac{2996598732849334755125678209477525144027947562227519372145080490021}{112329435833466175567732857751716368694755353295420466112682041029919} a^{2} - \frac{117070128192995101242045272862469641458497848476613649193806427889175}{1347953230001594106812794293020596424337064239545045593352184492359028} a - \frac{505111684080111362805198222407017405143520547825534368972081153421765}{1347953230001594106812794293020596424337064239545045593352184492359028}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $35$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 361611812727349340000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_{12}$ (as 36T3):

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 $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.2, 3.3.3969.1, 4.4.18432.1, 6.6.3359232.1, 6.6.1229312.1, 6.6.8065516032.2, 6.6.8065516032.1, 9.9.62523502209.1, 12.12.3327916660110655488.1, 12.12.36099543110378323968.1, 12.12.19184777290122566867877888.3, 12.12.19184777290122566867877888.1, 18.18.524682375772545974113841184768.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.12.33.376$x^{12} + 36 x^{10} + 42 x^{8} - 40 x^{6} + 40 x^{4} + 32 x^{2} - 56$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
2.12.33.376$x^{12} + 36 x^{10} + 42 x^{8} - 40 x^{6} + 40 x^{4} + 32 x^{2} - 56$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
2.12.33.376$x^{12} + 36 x^{10} + 42 x^{8} - 40 x^{6} + 40 x^{4} + 32 x^{2} - 56$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$