Properties

Label 45.15.174...691.1
Degree $45$
Signature $[15, 15]$
Discriminant $-1.744\times 10^{69}$
Root discriminant $34.57$
Ramified primes $3, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{15}\times S_3$ (as 45T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^45 - 12*x^44 + 42*x^43 + 36*x^42 - 525*x^41 + 702*x^40 + 2050*x^39 - 5802*x^38 - 2832*x^37 + 26919*x^36 - 20811*x^35 - 75546*x^34 + 201699*x^33 - 97125*x^32 - 532302*x^31 + 1289591*x^30 - 477339*x^29 - 2405610*x^28 + 4128988*x^27 - 1666824*x^26 - 5259345*x^25 + 9273382*x^24 + 3749217*x^23 - 15223818*x^22 - 9661128*x^21 + 33692574*x^20 - 4282803*x^19 - 38096273*x^18 + 36480543*x^17 - 14154921*x^16 + 10925288*x^15 - 437979*x^14 - 36804246*x^13 + 57064132*x^12 - 25852146*x^11 - 34300866*x^10 + 48986404*x^9 - 2084457*x^8 - 23365995*x^7 + 5433983*x^6 + 5348430*x^5 - 1230861*x^4 - 665734*x^3 + 68376*x^2 + 38973*x + 2969)
 
gp: K = bnfinit(x^45 - 12*x^44 + 42*x^43 + 36*x^42 - 525*x^41 + 702*x^40 + 2050*x^39 - 5802*x^38 - 2832*x^37 + 26919*x^36 - 20811*x^35 - 75546*x^34 + 201699*x^33 - 97125*x^32 - 532302*x^31 + 1289591*x^30 - 477339*x^29 - 2405610*x^28 + 4128988*x^27 - 1666824*x^26 - 5259345*x^25 + 9273382*x^24 + 3749217*x^23 - 15223818*x^22 - 9661128*x^21 + 33692574*x^20 - 4282803*x^19 - 38096273*x^18 + 36480543*x^17 - 14154921*x^16 + 10925288*x^15 - 437979*x^14 - 36804246*x^13 + 57064132*x^12 - 25852146*x^11 - 34300866*x^10 + 48986404*x^9 - 2084457*x^8 - 23365995*x^7 + 5433983*x^6 + 5348430*x^5 - 1230861*x^4 - 665734*x^3 + 68376*x^2 + 38973*x + 2969, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2969, 38973, 68376, -665734, -1230861, 5348430, 5433983, -23365995, -2084457, 48986404, -34300866, -25852146, 57064132, -36804246, -437979, 10925288, -14154921, 36480543, -38096273, -4282803, 33692574, -9661128, -15223818, 3749217, 9273382, -5259345, -1666824, 4128988, -2405610, -477339, 1289591, -532302, -97125, 201699, -75546, -20811, 26919, -2832, -5802, 2050, 702, -525, 36, 42, -12, 1]);
 

\( x^{45} - 12 x^{44} + 42 x^{43} + 36 x^{42} - 525 x^{41} + 702 x^{40} + 2050 x^{39} - 5802 x^{38} - 2832 x^{37} + 26919 x^{36} - 20811 x^{35} - 75546 x^{34} + 201699 x^{33} - 97125 x^{32} - 532302 x^{31} + 1289591 x^{30} - 477339 x^{29} - 2405610 x^{28} + 4128988 x^{27} - 1666824 x^{26} - 5259345 x^{25} + 9273382 x^{24} + 3749217 x^{23} - 15223818 x^{22} - 9661128 x^{21} + 33692574 x^{20} - 4282803 x^{19} - 38096273 x^{18} + 36480543 x^{17} - 14154921 x^{16} + 10925288 x^{15} - 437979 x^{14} - 36804246 x^{13} + 57064132 x^{12} - 25852146 x^{11} - 34300866 x^{10} + 48986404 x^{9} - 2084457 x^{8} - 23365995 x^{7} + 5433983 x^{6} + 5348430 x^{5} - 1230861 x^{4} - 665734 x^{3} + 68376 x^{2} + 38973 x + 2969 \)

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

Invariants

Degree:  $45$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[15, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-17\!\cdots\!691\)\(\medspace = -\,3^{60}\cdot 11^{39}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $34.57$
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:  $3, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $15$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $\frac{1}{89} a^{35} + \frac{40}{89} a^{34} + \frac{41}{89} a^{33} + \frac{5}{89} a^{32} - \frac{36}{89} a^{31} - \frac{11}{89} a^{30} - \frac{28}{89} a^{29} + \frac{33}{89} a^{28} + \frac{43}{89} a^{27} + \frac{25}{89} a^{25} - \frac{35}{89} a^{24} + \frac{21}{89} a^{23} + \frac{25}{89} a^{22} + \frac{21}{89} a^{21} + \frac{34}{89} a^{20} - \frac{33}{89} a^{19} + \frac{3}{89} a^{18} - \frac{27}{89} a^{17} - \frac{21}{89} a^{16} + \frac{10}{89} a^{15} + \frac{7}{89} a^{14} + \frac{11}{89} a^{13} - \frac{10}{89} a^{12} - \frac{4}{89} a^{11} - \frac{18}{89} a^{10} + \frac{4}{89} a^{9} - \frac{43}{89} a^{8} - \frac{34}{89} a^{7} + \frac{1}{89} a^{6} + \frac{11}{89} a^{5} - \frac{35}{89} a^{4} + \frac{44}{89} a^{3} - \frac{40}{89} a^{2} + \frac{6}{89} a + \frac{15}{89}$, $\frac{1}{89} a^{36} + \frac{43}{89} a^{34} - \frac{33}{89} a^{33} + \frac{31}{89} a^{32} + \frac{5}{89} a^{31} - \frac{33}{89} a^{30} - \frac{4}{89} a^{29} - \frac{31}{89} a^{28} - \frac{29}{89} a^{27} + \frac{25}{89} a^{26} + \frac{33}{89} a^{25} - \frac{3}{89} a^{24} - \frac{14}{89} a^{23} - \frac{5}{89} a^{21} + \frac{31}{89} a^{20} - \frac{12}{89} a^{19} + \frac{31}{89} a^{18} - \frac{9}{89} a^{17} - \frac{40}{89} a^{16} - \frac{37}{89} a^{15} - \frac{2}{89} a^{14} - \frac{5}{89} a^{13} + \frac{40}{89} a^{12} - \frac{36}{89} a^{11} + \frac{12}{89} a^{10} - \frac{25}{89} a^{9} - \frac{5}{89} a^{8} + \frac{26}{89} a^{7} - \frac{29}{89} a^{6} - \frac{30}{89} a^{5} + \frac{20}{89} a^{4} - \frac{20}{89} a^{3} + \frac{4}{89} a^{2} + \frac{42}{89} a + \frac{23}{89}$, $\frac{1}{89} a^{37} + \frac{27}{89} a^{34} - \frac{41}{89} a^{33} - \frac{32}{89} a^{32} + \frac{2}{89} a^{31} + \frac{24}{89} a^{30} + \frac{16}{89} a^{29} - \frac{24}{89} a^{28} - \frac{44}{89} a^{27} + \frac{33}{89} a^{26} - \frac{10}{89} a^{25} - \frac{22}{89} a^{24} - \frac{13}{89} a^{23} - \frac{12}{89} a^{22} + \frac{18}{89} a^{21} + \frac{39}{89} a^{20} + \frac{26}{89} a^{19} + \frac{40}{89} a^{18} - \frac{36}{89} a^{17} - \frac{24}{89} a^{16} + \frac{13}{89} a^{15} - \frac{39}{89} a^{14} + \frac{12}{89} a^{13} + \frac{38}{89} a^{12} + \frac{6}{89} a^{11} + \frac{37}{89} a^{10} + \frac{1}{89} a^{9} + \frac{6}{89} a^{8} + \frac{9}{89} a^{7} + \frac{16}{89} a^{6} - \frac{8}{89} a^{5} - \frac{28}{89} a^{4} - \frac{19}{89} a^{3} - \frac{18}{89} a^{2} + \frac{32}{89} a - \frac{22}{89}$, $\frac{1}{89} a^{38} + \frac{36}{89} a^{34} + \frac{18}{89} a^{33} - \frac{44}{89} a^{32} + \frac{17}{89} a^{31} - \frac{43}{89} a^{30} + \frac{20}{89} a^{29} + \frac{44}{89} a^{28} + \frac{29}{89} a^{27} - \frac{10}{89} a^{26} + \frac{15}{89} a^{25} + \frac{42}{89} a^{24} + \frac{44}{89} a^{23} - \frac{34}{89} a^{22} + \frac{6}{89} a^{21} - \frac{2}{89} a^{20} + \frac{41}{89} a^{19} - \frac{28}{89} a^{18} - \frac{7}{89} a^{17} - \frac{43}{89} a^{16} - \frac{42}{89} a^{15} + \frac{1}{89} a^{14} + \frac{8}{89} a^{13} + \frac{9}{89} a^{12} - \frac{33}{89} a^{11} + \frac{42}{89} a^{10} - \frac{13}{89} a^{9} + \frac{13}{89} a^{8} + \frac{44}{89} a^{7} - \frac{35}{89} a^{6} + \frac{31}{89} a^{5} + \frac{36}{89} a^{4} + \frac{40}{89} a^{3} + \frac{44}{89} a^{2} - \frac{6}{89} a + \frac{40}{89}$, $\frac{1}{89} a^{39} + \frac{2}{89} a^{34} - \frac{7}{89} a^{33} + \frac{15}{89} a^{32} + \frac{7}{89} a^{31} - \frac{29}{89} a^{30} - \frac{16}{89} a^{29} - \frac{2}{89} a^{28} + \frac{44}{89} a^{27} + \frac{15}{89} a^{26} + \frac{32}{89} a^{25} - \frac{31}{89} a^{24} + \frac{11}{89} a^{23} - \frac{4}{89} a^{22} + \frac{43}{89} a^{21} - \frac{26}{89} a^{20} + \frac{3}{89} a^{19} - \frac{26}{89} a^{18} + \frac{39}{89} a^{17} + \frac{2}{89} a^{16} - \frac{3}{89} a^{15} + \frac{23}{89} a^{14} - \frac{31}{89} a^{13} - \frac{29}{89} a^{12} + \frac{8}{89} a^{11} + \frac{12}{89} a^{10} - \frac{42}{89} a^{9} - \frac{10}{89} a^{8} + \frac{32}{89} a^{7} - \frac{5}{89} a^{6} - \frac{4}{89} a^{5} - \frac{35}{89} a^{4} - \frac{27}{89} a^{3} + \frac{10}{89} a^{2} + \frac{2}{89} a - \frac{6}{89}$, $\frac{1}{89} a^{40} + \frac{2}{89} a^{34} + \frac{22}{89} a^{33} - \frac{3}{89} a^{32} + \frac{43}{89} a^{31} + \frac{6}{89} a^{30} - \frac{35}{89} a^{29} - \frac{22}{89} a^{28} + \frac{18}{89} a^{27} + \frac{32}{89} a^{26} + \frac{8}{89} a^{25} - \frac{8}{89} a^{24} + \frac{43}{89} a^{23} - \frac{7}{89} a^{22} + \frac{21}{89} a^{21} + \frac{24}{89} a^{20} + \frac{40}{89} a^{19} + \frac{33}{89} a^{18} - \frac{33}{89} a^{17} + \frac{39}{89} a^{16} + \frac{3}{89} a^{15} + \frac{44}{89} a^{14} + \frac{38}{89} a^{13} + \frac{28}{89} a^{12} + \frac{20}{89} a^{11} - \frac{6}{89} a^{10} - \frac{18}{89} a^{9} + \frac{29}{89} a^{8} - \frac{26}{89} a^{7} - \frac{6}{89} a^{6} + \frac{32}{89} a^{5} + \frac{43}{89} a^{4} + \frac{11}{89} a^{3} - \frac{7}{89} a^{2} - \frac{18}{89} a - \frac{30}{89}$, $\frac{1}{89} a^{41} + \frac{31}{89} a^{34} + \frac{4}{89} a^{33} + \frac{33}{89} a^{32} - \frac{11}{89} a^{31} - \frac{13}{89} a^{30} + \frac{34}{89} a^{29} + \frac{41}{89} a^{28} + \frac{35}{89} a^{27} + \frac{8}{89} a^{26} + \frac{31}{89} a^{25} + \frac{24}{89} a^{24} + \frac{40}{89} a^{23} - \frac{29}{89} a^{22} - \frac{18}{89} a^{21} - \frac{28}{89} a^{20} + \frac{10}{89} a^{19} - \frac{39}{89} a^{18} + \frac{4}{89} a^{17} - \frac{44}{89} a^{16} + \frac{24}{89} a^{15} + \frac{24}{89} a^{14} + \frac{6}{89} a^{13} + \frac{40}{89} a^{12} + \frac{2}{89} a^{11} + \frac{18}{89} a^{10} + \frac{21}{89} a^{9} - \frac{29}{89} a^{8} - \frac{27}{89} a^{7} + \frac{30}{89} a^{6} + \frac{21}{89} a^{5} - \frac{8}{89} a^{4} - \frac{6}{89} a^{3} - \frac{27}{89} a^{2} - \frac{42}{89} a - \frac{30}{89}$, $\frac{1}{89} a^{42} + \frac{10}{89} a^{34} + \frac{8}{89} a^{33} + \frac{12}{89} a^{32} + \frac{35}{89} a^{31} + \frac{19}{89} a^{30} + \frac{19}{89} a^{29} - \frac{9}{89} a^{28} + \frac{10}{89} a^{27} + \frac{31}{89} a^{26} - \frac{39}{89} a^{25} - \frac{32}{89} a^{24} + \frac{32}{89} a^{23} + \frac{8}{89} a^{22} + \frac{33}{89} a^{21} + \frac{24}{89} a^{20} + \frac{5}{89} a^{19} - \frac{8}{89} a^{17} - \frac{37}{89} a^{16} - \frac{19}{89} a^{15} - \frac{33}{89} a^{14} - \frac{34}{89} a^{13} - \frac{44}{89} a^{12} - \frac{36}{89} a^{11} - \frac{44}{89} a^{10} + \frac{25}{89} a^{9} - \frac{29}{89} a^{8} + \frac{16}{89} a^{7} - \frac{10}{89} a^{6} + \frac{7}{89} a^{5} + \frac{11}{89} a^{4} + \frac{33}{89} a^{3} + \frac{41}{89} a^{2} - \frac{38}{89} a - \frac{20}{89}$, $\frac{1}{88199} a^{43} - \frac{290}{88199} a^{42} - \frac{15}{88199} a^{41} + \frac{53}{88199} a^{40} - \frac{250}{88199} a^{39} + \frac{125}{88199} a^{38} + \frac{421}{88199} a^{37} - \frac{165}{88199} a^{36} - \frac{49}{88199} a^{35} - \frac{12380}{88199} a^{34} - \frac{30750}{88199} a^{33} - \frac{38817}{88199} a^{32} + \frac{14409}{88199} a^{31} - \frac{22060}{88199} a^{30} + \frac{10957}{88199} a^{29} - \frac{9620}{88199} a^{28} - \frac{25735}{88199} a^{27} - \frac{33657}{88199} a^{26} + \frac{36079}{88199} a^{25} - \frac{25936}{88199} a^{24} - \frac{30249}{88199} a^{23} - \frac{29889}{88199} a^{22} - \frac{37165}{88199} a^{21} + \frac{30488}{88199} a^{20} - \frac{41108}{88199} a^{19} + \frac{25596}{88199} a^{18} - \frac{9502}{88199} a^{17} - \frac{42673}{88199} a^{16} + \frac{17371}{88199} a^{15} - \frac{33581}{88199} a^{14} + \frac{11033}{88199} a^{13} + \frac{9632}{88199} a^{12} + \frac{35096}{88199} a^{11} - \frac{26786}{88199} a^{10} + \frac{29557}{88199} a^{9} - \frac{29696}{88199} a^{8} + \frac{20438}{88199} a^{7} + \frac{16172}{88199} a^{6} - \frac{18860}{88199} a^{5} - \frac{16284}{88199} a^{4} + \frac{761}{88199} a^{3} + \frac{6766}{88199} a^{2} - \frac{8505}{88199} a + \frac{4872}{88199}$, $\frac{1}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{44} - \frac{48725853414312701088011661450144320355140568927266193002107325958168523607501449833774438597026781490926559090931230144889168028621974425070352707681360629876491}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{43} + \frac{20245359914126458120475002684371092024048155101084333475708620898704647146607263583501215101926551085237731094376376669270014188529230151934387680321993119550106361}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{42} + \frac{74389018566208251606473532139256569703465285545288007949122692110893257524170626070930938072461421887387500349583263735839714757120683098948691461977718776399454166}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{41} + \frac{54369612398802421027140778683160460291441749715077369232082106069831282894238416904256573060799442354500407194707075866708762964598618750887991928280620813923825957}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{40} - \frac{70116112007196853218143183523505866019959583098639853566945511981457634124891957283605495158220978685894120278602587009651271423989739681930299045322530591198387561}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{39} + \frac{58923332208356948727532353855973082101617810161854129613139131624804940775040073068668535982744996515897233481967855042363486639590432421480419815143188847855322400}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{38} + \frac{55506303052175855525005775474773575362747158970083692996473450688590026785213565047598553782317526245265740368167116196696168801031761282054326206640935936848416798}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{37} + \frac{42138499928530863887820932509495709870390959664495170560505411815346879218337239333246347738200920908389969830653401331041264945482366403598049911105522692547731651}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{36} - \frac{73727617782576397421917353027778168930256481377009237129120682109296852733711671765312729521437337673113583046553241457823241115245930984660027782816627680752760564}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{35} + \frac{3415445886904401680011758968889200093889822602067226388360235217752768694694882619884951386824580405220108059583985851194150764325701140495397700743796601939317576083}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{34} - \frac{6160490334749626933368604709939459547489973271866615801176972411204800710475417055168317627700970032043364098540839175389588573081982038929227027277134390600677801725}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{33} + \frac{6879934898454805103157885421741121098035174636421748542518857903013957603796045036268847014434504421915735916502577322462320739281116967467405471890957118851372017443}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{32} - \frac{4498953089215315977880265339915574477355801190648430538715469722872583761993978301487894335744402398612353907470344513804263135655272043478210096611576320335150676963}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{31} - \frac{1919608517143922333554574319360950986096507339845602852433404746266937449405465898589578351953944597749526128278943419961144711736920679488196034007608925547147315961}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{30} - \frac{5881944412438563001451925418208734701742251652470780965658164897041724798713410797469412953529813827711711471594000563178460692713479141489405983014283389706350742681}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{29} + \frac{6793432528075806185181765576372443414265971878437636454480525308518958527483397301516197444143312091936588030222859518691869401068695765328277591696254013320314212650}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{28} - \frac{4176787464839613493258778423033747654857949778536911694315104346609341308565758237669925826756910053113523031019204998358414745668178346326316966177447979887477465830}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{27} - \frac{1542586312133908179362529473707521269718476751882446985600272987766794832094481151192796164893469582618588256111995405441721818927258454699390475953426777532996306360}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{26} - \frac{2807617102637402608577494257402217741545960866671452466472049666040709698657339017538866585752299217914271630707507700133985930507439180600454229113214831822305995269}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{25} + \frac{2133355653754862267331988247196771248100365467994205233117026939790113656617372474412498642348494867931486591809030606076984466831590461725767689120073853631962761980}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{24} - \frac{171128393908594707772414407617914993838032529962407996361079190241005330976437604495682451442437019740502966371323360447933575283674129055277821739815139698977800356}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{23} - \frac{4731460227240042274552492695804048702763901410710484034910756877631082861932625335434342938424584288833403053455063899505884743174203540005989255695717218961296616207}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{22} + \frac{6045723158750320747500403439232365829381365177013060667216438156922485666669044599032880459029623721434134611871759750888288333053287582133605177882022116638514254935}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{21} + \frac{4696204991345197079352833725903449509079536286529907457506332312170668978600641201806696390454271566036498160911143395082489213934177422466939915026388887329529924947}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{20} - \frac{9415644154408281132085305749113630256594930540236271454960573623643676941143472342420725378384458796892379685701307922300409027198382407841648570972625281905414284}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{19} - \frac{1831297614824343532457567933204571421059904654202959815181792600357779920206457129268137936399632206240676825375922593067508459223704530587670865706168724787455308438}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{18} - \frac{2496125318711433794571089509153614240905781704546335126340358810929030744479411557848695798978455438616036026892079201587708876436649089346832788990313453900088522474}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{17} + \frac{1486686058457420415218182050236494020526821155227321858216909786210246506620704464216455905448571774338131955821369076620393397053758077366939588683113401682780900823}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{16} + \frac{1573408988703138500500005596473951941345916644840426882695316271686219694360469707215905762662815886408536820614092019510875360077243673784535504554633542268545459171}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{15} - \frac{2929467826924025603661223927438091485992105115811804368321249172273933382042646610343875962347464102384621519801288734138936058329395532934008700244531341965454003613}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{14} + \frac{1735034249910457125789134981084297501478116345997187302418185039238151656212645983668617126057146134022365768243067617643034431457513004002720732718341178843359069466}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{13} - \frac{3674674151686902782599547878091320874290542429365889195265724571876212117155070597387420431640928185007903999369456493505739046814203366375883023545145238030468810920}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{12} + \frac{1376137360275171207185772896703296334197167536393933815717080920705194583305903215468005051402919474412921486588104516878432119220716909712653956600804022954305773322}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{11} - \frac{4405021072008947900477969332123612094267607950402768574535352746192728866871360719340209491965290766683745031983498410418419194913544331167956952697037644004065985719}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{10} + \frac{5048524956454769999120459460303165049066604662357944318069690249927007550304039807910886552163532220084569379218122123921010669744066025267491933604362370801935824201}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{9} + \frac{794587213718624351824025485913110474672124191099303958594690369351072735002667240367384415766201692974181043873423303847502234093069271305202975095905981772053248302}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{8} + \frac{549046495050726125396450963490975225226031355901981382410576480506736469829855545281361814708469680564696756166937199017974394553967838406918190593608948017769334431}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{7} - \frac{785064283710802937143790857509384166823424157437131854579427755190972792531286868289916393415000082173681695927416031199527126356844614611324380425637208927179928101}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{6} + \frac{1690724211993895523168969219038419489476573989475199107180958916479150486367868896444304875385632782867771484657213733976165051663572626010159349249140344439705457748}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{5} + \frac{4929180107369766940958699281421042504457732377766371942648161982401863908881195744638953439638502993635693829923132410530441976542245166572916585459479650994052961122}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{4} + \frac{2838662383215932014206176457323180599409868952665653292054155237159336748326455656207439652305545539628014632056909250765025403795430697594687430323021594388791464497}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{3} + \frac{1612134525955238052390697688775172528628605144731876434064262974894785943912842490017978686143637528279785837563778623177791533821498285413122572739579572148121659548}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a^{2} - \frac{2450042845767484464169280796239464328336382157991366446853582101809869368613475786804525644171780052922944487632659321252371651672909987485145361687238317249525462244}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269} a + \frac{2917632680993283769783720919574662987479555095139798121927847129897659475586157934104120716860913967961594480015850830332101417038252600581599901196723323220841877458}{13790238882456353519886579580042742034578161838422515380022497377058171621099306340244517322238907055420514091770697959992870100552537176213595854735894426340001459269}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $29$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 495912260998404300 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{15}\cdot(2\pi)^{15}\cdot 495912260998404300 \cdot 1}{2\sqrt{1744174787462733669236743003323349838789896331385003649618101010724691}}\approx 0.182695482397487$ (assuming GRH)

Galois group

$C_{15}\times S_3$ (as 45T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 90
The 45 conjugacy class representatives for $C_{15}\times S_3$
Character table for $C_{15}\times S_3$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.1.891.1, \(\Q(\zeta_{11})^+\), 9.3.707347971.1, 15.15.10943023107606534329121.1, 15.5.120373254183671877620331.1

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 $30{,}\,15$ R $15^{3}$ $30{,}\,15$ R $30{,}\,15$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{3}{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{15}$ $30{,}\,15$ $15^{3}$ $15^{3}$ $30{,}\,15$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{5}$ $15^{3}$ $15^{3}$ $15^{3}$

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
3Data not computed
11Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.11.2t1.a.a$1$ $ 11 $ \(\Q(\sqrt{-11}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.99.6t1.a.a$1$ $ 3^{2} \cdot 11 $ 6.0.8732691.1 $C_6$ (as 6T1) $0$ $-1$
1.99.6t1.a.b$1$ $ 3^{2} \cdot 11 $ 6.0.8732691.1 $C_6$ (as 6T1) $0$ $-1$
* 1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.11.5t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.5t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.11.10t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
1.11.10t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
1.11.10t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.11.10t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.99.15t1.a.a$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.99.15t1.a.b$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
1.99.30t1.a.a$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
1.99.30t1.a.b$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
1.99.30t1.a.c$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
* 1.99.15t1.a.c$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.99.15t1.a.d$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.99.15t1.a.e$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.99.15t1.a.f$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
1.99.30t1.a.d$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
1.99.30t1.a.e$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
* 1.99.15t1.a.g$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
1.99.30t1.a.f$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
1.99.30t1.a.g$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
* 1.99.15t1.a.h$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
1.99.30t1.a.h$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
* 2.891.3t2.b.a$2$ $ 3^{4} \cdot 11 $ 3.1.891.1 $S_3$ (as 3T2) $1$ $0$
* 2.99.6t5.a.a$2$ $ 3^{2} \cdot 11 $ 6.0.107811.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.99.6t5.a.b$2$ $ 3^{2} \cdot 11 $ 6.0.107811.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.9801.15t4.a.a$2$ $ 3^{4} \cdot 11^{2}$ 15.5.120373254183671877620331.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.9801.15t4.a.b$2$ $ 3^{4} \cdot 11^{2}$ 15.5.120373254183671877620331.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.9801.15t4.a.c$2$ $ 3^{4} \cdot 11^{2}$ 15.5.120373254183671877620331.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.9801.15t4.a.d$2$ $ 3^{4} \cdot 11^{2}$ 15.5.120373254183671877620331.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.1089.30t15.b.a$2$ $ 3^{2} \cdot 11^{2}$ 45.15.1744174787462733669236743003323349838789896331385003649618101010724691.1 $C_{15}\times S_3$ (as 45T3) $0$ $0$
* 2.1089.30t15.b.b$2$ $ 3^{2} \cdot 11^{2}$ 45.15.1744174787462733669236743003323349838789896331385003649618101010724691.1 $C_{15}\times S_3$ (as 45T3) $0$ $0$
* 2.1089.30t15.b.c$2$ $ 3^{2} \cdot 11^{2}$ 45.15.1744174787462733669236743003323349838789896331385003649618101010724691.1 $C_{15}\times S_3$ (as 45T3) $0$ $0$
* 2.1089.30t15.b.d$2$ $ 3^{2} \cdot 11^{2}$ 45.15.1744174787462733669236743003323349838789896331385003649618101010724691.1 $C_{15}\times S_3$ (as 45T3) $0$ $0$
* 2.1089.30t15.b.e$2$ $ 3^{2} \cdot 11^{2}$ 45.15.1744174787462733669236743003323349838789896331385003649618101010724691.1 $C_{15}\times S_3$ (as 45T3) $0$ $0$
* 2.1089.30t15.b.f$2$ $ 3^{2} \cdot 11^{2}$ 45.15.1744174787462733669236743003323349838789896331385003649618101010724691.1 $C_{15}\times S_3$ (as 45T3) $0$ $0$
* 2.1089.30t15.b.g$2$ $ 3^{2} \cdot 11^{2}$ 45.15.1744174787462733669236743003323349838789896331385003649618101010724691.1 $C_{15}\times S_3$ (as 45T3) $0$ $0$
* 2.1089.30t15.b.h$2$ $ 3^{2} \cdot 11^{2}$ 45.15.1744174787462733669236743003323349838789896331385003649618101010724691.1 $C_{15}\times S_3$ (as 45T3) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.