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Properties

Label 36.0.42067205642...3125.1
Degree $36$
Signature $[0, 18]$
Discriminant $5^{18}\cdot 7^{24}\cdot 13^{33}$
Root discriminant $85.90$
Ramified primes $5, 7, 13$
Class number Not computed
Class group Not computed
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![32332243789, -20963341532, 47347720069, -71789180593, 123224165522, -135288523899, 186781470161, -177210083687, 175257385390, -131090685506, 98895340726, -57654424490, 33733591464, -15030668107, 7079450070, -2319599256, 928905669, -201850025, 93286907, -18096988, 16165015, -5937815, 4589322, -2161107, 1314509, -609219, 301106, -126998, 55175, -21073, 7774, -2473, 763, -208, 52, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 9*x^35 + 52*x^34 - 208*x^33 + 763*x^32 - 2473*x^31 + 7774*x^30 - 21073*x^29 + 55175*x^28 - 126998*x^27 + 301106*x^26 - 609219*x^25 + 1314509*x^24 - 2161107*x^23 + 4589322*x^22 - 5937815*x^21 + 16165015*x^20 - 18096988*x^19 + 93286907*x^18 - 201850025*x^17 + 928905669*x^16 - 2319599256*x^15 + 7079450070*x^14 - 15030668107*x^13 + 33733591464*x^12 - 57654424490*x^11 + 98895340726*x^10 - 131090685506*x^9 + 175257385390*x^8 - 177210083687*x^7 + 186781470161*x^6 - 135288523899*x^5 + 123224165522*x^4 - 71789180593*x^3 + 47347720069*x^2 - 20963341532*x + 32332243789)
 
gp: K = bnfinit(x^36 - 9*x^35 + 52*x^34 - 208*x^33 + 763*x^32 - 2473*x^31 + 7774*x^30 - 21073*x^29 + 55175*x^28 - 126998*x^27 + 301106*x^26 - 609219*x^25 + 1314509*x^24 - 2161107*x^23 + 4589322*x^22 - 5937815*x^21 + 16165015*x^20 - 18096988*x^19 + 93286907*x^18 - 201850025*x^17 + 928905669*x^16 - 2319599256*x^15 + 7079450070*x^14 - 15030668107*x^13 + 33733591464*x^12 - 57654424490*x^11 + 98895340726*x^10 - 131090685506*x^9 + 175257385390*x^8 - 177210083687*x^7 + 186781470161*x^6 - 135288523899*x^5 + 123224165522*x^4 - 71789180593*x^3 + 47347720069*x^2 - 20963341532*x + 32332243789, 1)
 

Normalized defining polynomial

\( x^{36} - 9 x^{35} + 52 x^{34} - 208 x^{33} + 763 x^{32} - 2473 x^{31} + 7774 x^{30} - 21073 x^{29} + 55175 x^{28} - 126998 x^{27} + 301106 x^{26} - 609219 x^{25} + 1314509 x^{24} - 2161107 x^{23} + 4589322 x^{22} - 5937815 x^{21} + 16165015 x^{20} - 18096988 x^{19} + 93286907 x^{18} - 201850025 x^{17} + 928905669 x^{16} - 2319599256 x^{15} + 7079450070 x^{14} - 15030668107 x^{13} + 33733591464 x^{12} - 57654424490 x^{11} + 98895340726 x^{10} - 131090685506 x^{9} + 175257385390 x^{8} - 177210083687 x^{7} + 186781470161 x^{6} - 135288523899 x^{5} + 123224165522 x^{4} - 71789180593 x^{3} + 47347720069 x^{2} - 20963341532 x + 32332243789 \)

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:  \(4206720564246829878857376609821276819265009722410575911914825439453125=5^{18}\cdot 7^{24}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.90$
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:  $5, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(455=5\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{455}(256,·)$, $\chi_{455}(1,·)$, $\chi_{455}(386,·)$, $\chi_{455}(261,·)$, $\chi_{455}(16,·)$, $\chi_{455}(149,·)$, $\chi_{455}(284,·)$, $\chi_{455}(414,·)$, $\chi_{455}(36,·)$, $\chi_{455}(296,·)$, $\chi_{455}(44,·)$, $\chi_{455}(51,·)$, $\chi_{455}(184,·)$, $\chi_{455}(186,·)$, $\chi_{455}(316,·)$, $\chi_{455}(191,·)$, $\chi_{455}(449,·)$, $\chi_{455}(326,·)$, $\chi_{455}(81,·)$, $\chi_{455}(211,·)$, $\chi_{455}(214,·)$, $\chi_{455}(121,·)$, $\chi_{455}(344,·)$, $\chi_{455}(219,·)$, $\chi_{455}(99,·)$, $\chi_{455}(359,·)$, $\chi_{455}(379,·)$, $\chi_{455}(361,·)$, $\chi_{455}(109,·)$, $\chi_{455}(239,·)$, $\chi_{455}(424,·)$, $\chi_{455}(116,·)$, $\chi_{455}(246,·)$, $\chi_{455}(249,·)$, $\chi_{455}(319,·)$, $\chi_{455}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{233} a^{21} + \frac{27}{233} a^{20} + \frac{109}{233} a^{19} - \frac{106}{233} a^{18} + \frac{88}{233} a^{17} + \frac{30}{233} a^{16} + \frac{31}{233} a^{15} - \frac{94}{233} a^{14} - \frac{49}{233} a^{13} + \frac{90}{233} a^{12} - \frac{40}{233} a^{11} - \frac{63}{233} a^{10} + \frac{103}{233} a^{9} + \frac{83}{233} a^{8} - \frac{58}{233} a^{7} - \frac{23}{233} a^{6} - \frac{95}{233} a^{5} + \frac{55}{233} a^{4} - \frac{69}{233} a^{3} - \frac{69}{233} a^{2} + \frac{6}{233} a - \frac{30}{233}$, $\frac{1}{233} a^{22} + \frac{79}{233} a^{20} - \frac{20}{233} a^{19} - \frac{79}{233} a^{18} - \frac{16}{233} a^{17} - \frac{80}{233} a^{16} + \frac{1}{233} a^{15} - \frac{74}{233} a^{14} + \frac{15}{233} a^{13} + \frac{93}{233} a^{12} + \frac{85}{233} a^{11} - \frac{60}{233} a^{10} + \frac{98}{233} a^{9} + \frac{31}{233} a^{8} - \frac{88}{233} a^{7} + \frac{60}{233} a^{6} + \frac{57}{233} a^{5} + \frac{77}{233} a^{4} - \frac{70}{233} a^{3} + \frac{5}{233} a^{2} + \frac{41}{233} a + \frac{111}{233}$, $\frac{1}{233} a^{23} - \frac{56}{233} a^{20} - \frac{69}{233} a^{19} - \frac{30}{233} a^{18} - \frac{42}{233} a^{17} - \frac{39}{233} a^{16} + \frac{40}{233} a^{15} - \frac{15}{233} a^{14} + \frac{3}{233} a^{13} - \frac{35}{233} a^{12} + \frac{71}{233} a^{11} - \frac{51}{233} a^{10} + \frac{49}{233} a^{9} + \frac{112}{233} a^{8} - \frac{18}{233} a^{7} + \frac{10}{233} a^{6} - \frac{107}{233} a^{5} + \frac{12}{233} a^{4} + \frac{97}{233} a^{3} - \frac{100}{233} a^{2} + \frac{103}{233} a + \frac{40}{233}$, $\frac{1}{233} a^{24} + \frac{45}{233} a^{20} + \frac{16}{233} a^{19} + \frac{80}{233} a^{18} - \frac{4}{233} a^{17} + \frac{89}{233} a^{16} + \frac{90}{233} a^{15} + \frac{98}{233} a^{14} + \frac{17}{233} a^{13} - \frac{15}{233} a^{12} + \frac{39}{233} a^{11} + \frac{16}{233} a^{10} + \frac{55}{233} a^{9} - \frac{30}{233} a^{8} + \frac{24}{233} a^{7} + \frac{3}{233} a^{6} + \frac{51}{233} a^{5} - \frac{85}{233} a^{4} - \frac{3}{233} a^{3} - \frac{33}{233} a^{2} - \frac{90}{233} a - \frac{49}{233}$, $\frac{1}{233} a^{25} - \frac{34}{233} a^{20} + \frac{68}{233} a^{19} + \frac{106}{233} a^{18} + \frac{90}{233} a^{17} - \frac{95}{233} a^{16} + \frac{101}{233} a^{15} + \frac{53}{233} a^{14} + \frac{93}{233} a^{13} - \frac{50}{233} a^{12} - \frac{48}{233} a^{11} + \frac{94}{233} a^{10} - \frac{5}{233} a^{9} + \frac{17}{233} a^{8} + \frac{50}{233} a^{7} - \frac{79}{233} a^{6} - \frac{4}{233} a^{5} + \frac{85}{233} a^{4} + \frac{43}{233} a^{3} - \frac{14}{233} a^{2} - \frac{86}{233} a - \frac{48}{233}$, $\frac{1}{233} a^{26} + \frac{54}{233} a^{20} + \frac{84}{233} a^{19} - \frac{19}{233} a^{18} + \frac{101}{233} a^{17} - \frac{44}{233} a^{16} - \frac{58}{233} a^{15} - \frac{74}{233} a^{14} - \frac{85}{233} a^{13} - \frac{17}{233} a^{12} - \frac{101}{233} a^{11} - \frac{50}{233} a^{10} + \frac{24}{233} a^{9} + \frac{76}{233} a^{8} + \frac{46}{233} a^{7} - \frac{87}{233} a^{6} - \frac{116}{233} a^{5} + \frac{49}{233} a^{4} - \frac{30}{233} a^{3} - \frac{102}{233} a^{2} - \frac{77}{233} a - \frac{88}{233}$, $\frac{1}{233} a^{27} + \frac{24}{233} a^{20} - \frac{80}{233} a^{19} + \frac{97}{233} a^{17} - \frac{47}{233} a^{16} + \frac{116}{233} a^{15} + \frac{98}{233} a^{14} + \frac{66}{233} a^{13} - \frac{68}{233} a^{12} + \frac{13}{233} a^{11} - \frac{69}{233} a^{10} + \frac{106}{233} a^{9} - \frac{9}{233} a^{8} + \frac{16}{233} a^{7} - \frac{39}{233} a^{6} + \frac{53}{233} a^{5} + \frac{29}{233} a^{4} - \frac{104}{233} a^{3} - \frac{79}{233} a^{2} + \frac{54}{233} a - \frac{11}{233}$, $\frac{1}{233} a^{28} - \frac{29}{233} a^{20} - \frac{53}{233} a^{19} + \frac{78}{233} a^{18} - \frac{62}{233} a^{17} + \frac{95}{233} a^{16} + \frac{53}{233} a^{15} - \frac{8}{233} a^{14} - \frac{57}{233} a^{13} - \frac{50}{233} a^{12} - \frac{41}{233} a^{11} - \frac{13}{233} a^{10} + \frac{82}{233} a^{9} - \frac{112}{233} a^{8} - \frac{45}{233} a^{7} - \frac{94}{233} a^{6} - \frac{21}{233} a^{5} - \frac{26}{233} a^{4} - \frac{54}{233} a^{3} + \frac{79}{233} a^{2} + \frac{78}{233} a + \frac{21}{233}$, $\frac{1}{233} a^{29} + \frac{31}{233} a^{20} - \frac{23}{233} a^{19} - \frac{107}{233} a^{18} + \frac{84}{233} a^{17} - \frac{9}{233} a^{16} - \frac{41}{233} a^{15} + \frac{13}{233} a^{14} - \frac{73}{233} a^{13} + \frac{6}{233} a^{12} - \frac{8}{233} a^{11} - \frac{114}{233} a^{10} + \frac{79}{233} a^{9} + \frac{32}{233} a^{8} + \frac{88}{233} a^{7} + \frac{11}{233} a^{6} + \frac{15}{233} a^{5} - \frac{90}{233} a^{4} - \frac{58}{233} a^{3} - \frac{59}{233} a^{2} - \frac{38}{233} a + \frac{62}{233}$, $\frac{1}{233} a^{30} + \frac{72}{233} a^{20} + \frac{9}{233} a^{19} + \frac{108}{233} a^{18} + \frac{59}{233} a^{17} - \frac{39}{233} a^{16} - \frac{16}{233} a^{15} + \frac{45}{233} a^{14} - \frac{106}{233} a^{13} - \frac{2}{233} a^{12} - \frac{39}{233} a^{11} - \frac{65}{233} a^{10} + \frac{101}{233} a^{9} + \frac{78}{233} a^{8} - \frac{55}{233} a^{7} + \frac{29}{233} a^{6} + \frac{59}{233} a^{5} + \frac{101}{233} a^{4} - \frac{17}{233} a^{3} + \frac{4}{233} a^{2} + \frac{109}{233} a - \frac{2}{233}$, $\frac{1}{233} a^{31} - \frac{71}{233} a^{20} - \frac{51}{233} a^{19} + \frac{2}{233} a^{18} - \frac{84}{233} a^{17} - \frac{79}{233} a^{16} - \frac{90}{233} a^{15} - \frac{95}{233} a^{14} + \frac{31}{233} a^{13} + \frac{5}{233} a^{12} + \frac{19}{233} a^{11} - \frac{23}{233} a^{10} - \frac{115}{233} a^{9} + \frac{27}{233} a^{8} + \frac{11}{233} a^{7} + \frac{84}{233} a^{6} - \frac{49}{233} a^{5} - \frac{16}{233} a^{4} + \frac{79}{233} a^{3} - \frac{49}{233} a^{2} + \frac{32}{233} a + \frac{63}{233}$, $\frac{1}{233} a^{32} + \frac{2}{233} a^{20} + \frac{52}{233} a^{19} + \frac{79}{233} a^{18} + \frac{111}{233} a^{17} - \frac{57}{233} a^{16} + \frac{9}{233} a^{15} + \frac{114}{233} a^{14} + \frac{21}{233} a^{13} - \frac{115}{233} a^{12} - \frac{67}{233} a^{11} + \frac{72}{233} a^{10} - \frac{116}{233} a^{9} + \frac{79}{233} a^{8} - \frac{73}{233} a^{7} - \frac{51}{233} a^{6} - \frac{4}{233} a^{5} + \frac{23}{233} a^{4} - \frac{55}{233} a^{3} + \frac{26}{233} a^{2} + \frac{23}{233} a - \frac{33}{233}$, $\frac{1}{233} a^{33} - \frac{2}{233} a^{20} + \frac{94}{233} a^{19} + \frac{90}{233} a^{18} - \frac{51}{233} a^{16} + \frac{52}{233} a^{15} - \frac{24}{233} a^{14} - \frac{17}{233} a^{13} - \frac{14}{233} a^{12} - \frac{81}{233} a^{11} + \frac{10}{233} a^{10} + \frac{106}{233} a^{9} - \frac{6}{233} a^{8} + \frac{65}{233} a^{7} + \frac{42}{233} a^{6} - \frac{20}{233} a^{5} + \frac{68}{233} a^{4} - \frac{69}{233} a^{3} - \frac{72}{233} a^{2} - \frac{45}{233} a + \frac{60}{233}$, $\frac{1}{69334043} a^{34} + \frac{52882}{69334043} a^{33} - \frac{15380}{69334043} a^{32} + \frac{69488}{69334043} a^{31} + \frac{5102}{69334043} a^{30} - \frac{136593}{69334043} a^{29} + \frac{59925}{69334043} a^{28} - \frac{9084}{69334043} a^{27} - \frac{117201}{69334043} a^{26} + \frac{37282}{69334043} a^{25} - \frac{48668}{69334043} a^{24} - \frac{65375}{69334043} a^{23} - \frac{143468}{69334043} a^{22} - \frac{39936}{69334043} a^{21} - \frac{23555555}{69334043} a^{20} - \frac{7283099}{69334043} a^{19} - \frac{23211544}{69334043} a^{18} - \frac{24655769}{69334043} a^{17} + \frac{6103381}{69334043} a^{16} + \frac{18609785}{69334043} a^{15} - \frac{32505238}{69334043} a^{14} - \frac{20893403}{69334043} a^{13} + \frac{20900272}{69334043} a^{12} + \frac{15495936}{69334043} a^{11} + \frac{24966489}{69334043} a^{10} + \frac{32329271}{69334043} a^{9} - \frac{3378111}{69334043} a^{8} - \frac{26486068}{69334043} a^{7} + \frac{6620099}{69334043} a^{6} + \frac{14279621}{69334043} a^{5} + \frac{11091872}{69334043} a^{4} + \frac{2346780}{69334043} a^{3} - \frac{8365108}{69334043} a^{2} + \frac{34434287}{69334043} a - \frac{5734519}{69334043}$, $\frac{1}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{35} - \frac{13974996220412881683267415088694659744534747896285402370755179215052757368221095310048509255335856507900401014002135891819715086566149136233668461331429020353159687610}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{34} + \frac{4332747837602768437994452127730916847759908942475467597107269617163775261342894595625014694290810802004339450830521786466206528698561214296659816968234173919316036717136099}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{33} + \frac{3210082499424872014565789724939386558593016460875649415494893327717794145425025803511819902161365787619210536573161942464986453444385687512490818555031118029168367781913093}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{32} - \frac{188753743714819716200898771739292233209403624852362863923257789718750982882506756720238901301908280551665426357429983181535163071662820751955787168249041816210707400919220}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{31} - \frac{1429047070454757621392903245611633763748890833174880620928816672471726875035019937487396435077422260777390170990781916988413640489865342543799615810451376128810233169028752}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{30} + \frac{3065703816713870139781075567820639230240797453384329375180689276998591194951647059186193315621222601797882889631878009591819779477253213181761954808669219325436634686356551}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{29} - \frac{385872921192786087254977777052546202079477408633613738457676055207421797661241822556503163639542960402126794149548526986344638956391436196248889744298498058854947438033650}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{28} - \frac{2702703218394844869985490137123091796304596881618299040768907002468180995006814288788498859755259198818098093193661165254427542772137747460261835006647576151208418469823769}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{27} - \frac{2399932098195122732525685419439843750449577495398222619552356852236935911654623929705494863710528557708634892796591095583035591992278170902503759344809433778316002795991204}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{26} - \frac{1240927784437205194293962391545667748183466478640112024865016507675149032991961416989197076550504281956155741383912726310573802129389825119917267792576224172319831158264685}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{25} + \frac{5605688481361815523131794869067754151661489030565776503731366355536609457231223977707064005911551899059739025316559295381478542392398442920366394051973235414917397175739381}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{24} + \frac{4702730467361442618289392270950817845747141764983489572379010830421286999867411060219995293921937512544243406692058291915039821834819567939594725811358153167563817732539016}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{23} + \frac{3946963026288814541064490197476345413177237971536748325463416373022139626533602078296890540761152547161194822271090619337403120635312096022296652854034741441264622317654605}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{22} + \frac{2341339883333048319614626375575576007309629805876671125775884122603417927906406416697573850294824195523040113805214659807717713474691282819338216285538972149485660841325321}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{21} + \frac{1146779608025952896381535139494664006974181058413325111714918813109018741235106342163642822997126245842276420351423752277929922497642522009393335675296209623611249666787738872}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{20} + \frac{1135179517321053732514681438003920915428304570593223011277109251232263488028432905040782180151534083325344806663700298217986651385518240344479274023963731193052771272071566965}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{19} + \frac{4180634618803122579445579837797616543106633081824995312449123743452391869498404452071845493723935973458856640043510065872686120596789403049286025667159019738536561806450205}{8390116585465831227513209131231752111814446938213318249762741248691389857034555105182612428764716611691032003140000778459614344298595422051631552358644083237584706432097789} a^{18} + \frac{1083152630207471804996080272447670556022985775875035132650524269581752955757107618727725293439284676084338935791509801252067998705637952471823971807313395203463761452601524696}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{17} + \frac{1403723141599050974648680953558601523961185957801210732527704893860690018569087981326567429477463262497137884253040142531130937125531477125782672634003485523157693977452543358}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{16} - \frac{468334830150970428952966700849257275285430845111649267379468303989106031150302156540085836700485292568625958675252506169921592850227349946295169353597929882326607673585718852}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{15} - \frac{324338597051954016944841008284351237045316191594080177564779981579264166765224277580966684046567775660214016239550416945233141235622969620240530675272317503625945632311598451}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{14} + \frac{25474186113527790225213337252910505641683836950237517980850230583390802373044562636283382757593285291039068910978548508383541678307523714315265356260277746415414685214416661}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{13} + \frac{119602807914730891431554037192804894061090655940428187247358044514907970640900506362577681439066001428188241645334821402034342199586600794661846600774617561300432312189957774}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{12} + \frac{1130518891687220200735337248347100679393624126990019493496275704943023396116524413293637850208446938423236827075607777589111750194831464518934515734126040833198803248506878845}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{11} - \frac{477506753270798616464072602362576612805584860077104473304655529225318480903183457546803821064787348437958972849484896751682688127384321963511267029873379363271272776991571674}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{10} - \frac{1081819311030406978796946985912831836925617965490365421637658549570161466028286269759005540481357029786544698841402332028717726902921148902856727681491074162738742010395163516}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{9} - \frac{838678679873224622117014037043919277569190360625877231786781134110350403910843596149929417104465027504392080419091326513497233233385475675010836264602458470039207324430884561}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{8} + \frac{749362700312256444607573565104513724194531766338418192844976283966811315935569801510101801111809316609028751344430733841175386253255332342578720593303646930467629743852200621}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{7} - \frac{772229445426355251698568950281269982410835490706243015986839617893985242038312959428178637356707870389875290092749571988545064806518976023327726533367551719185231219167605322}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{6} - \frac{762736398976593392716629884150924192384555755494301659534931920465641169406777978105920391036261283553282853834984547878023653364683618018197133677872614462792830487708884732}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{5} + \frac{450290059927294908798149884338585680730766412754467565946943687098712121269071653676128597817378464836490175186690333886544850308867393258510324401296714312631851937752633235}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{4} - \frac{820692368703061247067322842502494904915511520272309125700957051582070800391575133257549467794023706905807039063296089515620123875390611477145383736421111574972872051555696061}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{3} - \frac{194175274019874740064797197166424445499232774957483747457442298826220118562948831762543897167699309709426459647693815879154083660636289850408436822843375890237621095362589684}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a^{2} + \frac{599238828387991079048087469133250330422924519476038831106822033814290320550859326242315308040831908849007906697808012826350611649012091459168096622555745881776481083408601596}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893} a - \frac{703261163436054588431747738627126160105228891337860446132902274275439796035528139803881814691788257168935252418438191242573063839261574552651580408007374921361136430194644632}{2827469289301985123671951477225100461681468618177888250170043800808998381820645070446540388493709498139877785058180262340890034028626657231399833144863056051066046067616954893}$

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:  \( -1 \) (order $2$)
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

$C_3\times C_{12}$ (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

\(\Q(\sqrt{13}) \), 3.3.169.1, 3.3.8281.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 4.0.54925.1, \(\Q(\zeta_{13})^+\), 6.6.891474493.1, 6.6.891474493.2, 6.6.5274997.1, 9.9.567869252041.1, 12.0.28002506156828125.1, 12.0.161428875495388931828125.1, 12.0.161428875495388931828125.2, 12.0.955200446718277703125.1, 18.18.708478645847689707516501157.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$ R R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
$5$5.12.6.2$x^{12} - 3125 x^{2} + 31250$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
5.12.6.2$x^{12} - 3125 x^{2} + 31250$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
5.12.6.2$x^{12} - 3125 x^{2} + 31250$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
7Data not computed
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$