Properties

Label 30.0.182...875.1
Degree $30$
Signature $[0, 15]$
Discriminant $-1.820\times 10^{43}$
Root discriminant $27.67$
Ramified primes $3, 5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 10*x^27 - 54*x^25 + 90*x^24 + 315*x^23 + 15*x^22 - 240*x^21 + 504*x^20 - 4740*x^19 + 8375*x^18 - 14580*x^17 + 25140*x^16 - 34179*x^15 + 51465*x^14 - 61710*x^13 + 70215*x^12 - 67800*x^11 + 45276*x^10 - 22840*x^9 - 3240*x^8 + 16350*x^7 - 10730*x^6 + 4956*x^5 + 2280*x^4 - 4065*x^3 - 420*x^2 + 780*x + 169)
 
gp: K = bnfinit(x^30 - 10*x^27 - 54*x^25 + 90*x^24 + 315*x^23 + 15*x^22 - 240*x^21 + 504*x^20 - 4740*x^19 + 8375*x^18 - 14580*x^17 + 25140*x^16 - 34179*x^15 + 51465*x^14 - 61710*x^13 + 70215*x^12 - 67800*x^11 + 45276*x^10 - 22840*x^9 - 3240*x^8 + 16350*x^7 - 10730*x^6 + 4956*x^5 + 2280*x^4 - 4065*x^3 - 420*x^2 + 780*x + 169, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, 780, -420, -4065, 2280, 4956, -10730, 16350, -3240, -22840, 45276, -67800, 70215, -61710, 51465, -34179, 25140, -14580, 8375, -4740, 504, -240, 15, 315, 90, -54, 0, -10, 0, 0, 1]);
 

\( x^{30} - 10 x^{27} - 54 x^{25} + 90 x^{24} + 315 x^{23} + 15 x^{22} - 240 x^{21} + 504 x^{20} - 4740 x^{19} + 8375 x^{18} - 14580 x^{17} + 25140 x^{16} - 34179 x^{15} + 51465 x^{14} - 61710 x^{13} + 70215 x^{12} - 67800 x^{11} + 45276 x^{10} - 22840 x^{9} - 3240 x^{8} + 16350 x^{7} - 10730 x^{6} + 4956 x^{5} + 2280 x^{4} - 4065 x^{3} - 420 x^{2} + 780 x + 169 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-18201370075621419164235703647136688232421875\)\(\medspace = -\,3^{35}\cdot 5^{38}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $27.67$
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:  $3, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{5} a^{20} - \frac{1}{5} a^{15} + \frac{2}{5} a^{10} + \frac{1}{5} a^{5} + \frac{1}{5}$, $\frac{1}{5} a^{21} - \frac{1}{5} a^{16} + \frac{2}{5} a^{11} + \frac{1}{5} a^{6} + \frac{1}{5} a$, $\frac{1}{35} a^{22} + \frac{3}{35} a^{21} + \frac{2}{35} a^{20} - \frac{2}{7} a^{19} - \frac{16}{35} a^{17} + \frac{1}{5} a^{16} + \frac{3}{35} a^{15} - \frac{1}{7} a^{14} + \frac{2}{7} a^{13} - \frac{3}{35} a^{12} + \frac{11}{35} a^{11} + \frac{9}{35} a^{10} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{4}{35} a^{7} - \frac{12}{35} a^{6} + \frac{12}{35} a^{5} - \frac{3}{7} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{17}{35}$, $\frac{1}{35} a^{23} - \frac{2}{35} a^{20} - \frac{1}{7} a^{19} - \frac{16}{35} a^{18} - \frac{3}{7} a^{17} + \frac{2}{7} a^{16} + \frac{1}{5} a^{15} - \frac{2}{7} a^{14} + \frac{2}{35} a^{13} - \frac{3}{7} a^{12} - \frac{2}{7} a^{11} + \frac{6}{35} a^{10} + \frac{1}{7} a^{9} + \frac{6}{35} a^{8} - \frac{3}{7} a^{6} - \frac{2}{35} a^{5} + \frac{2}{7} a^{4} - \frac{2}{5} a^{3} + \frac{2}{7} a - \frac{2}{35}$, $\frac{1}{35} a^{24} - \frac{2}{35} a^{21} + \frac{2}{35} a^{20} - \frac{16}{35} a^{19} - \frac{3}{7} a^{18} + \frac{2}{7} a^{17} + \frac{1}{5} a^{16} - \frac{17}{35} a^{15} + \frac{2}{35} a^{14} - \frac{3}{7} a^{13} - \frac{2}{7} a^{12} + \frac{6}{35} a^{11} - \frac{16}{35} a^{10} + \frac{6}{35} a^{9} - \frac{3}{7} a^{7} - \frac{2}{35} a^{6} + \frac{17}{35} a^{5} - \frac{2}{5} a^{4} + \frac{2}{7} a^{2} - \frac{2}{35} a + \frac{1}{5}$, $\frac{1}{455} a^{25} + \frac{6}{455} a^{24} + \frac{1}{91} a^{22} + \frac{4}{455} a^{21} - \frac{11}{455} a^{20} + \frac{99}{455} a^{19} + \frac{40}{91} a^{18} + \frac{40}{91} a^{17} + \frac{116}{455} a^{16} + \frac{187}{455} a^{15} + \frac{172}{455} a^{14} - \frac{34}{91} a^{13} - \frac{36}{91} a^{12} + \frac{153}{455} a^{11} - \frac{139}{455} a^{10} + \frac{106}{455} a^{9} - \frac{38}{91} a^{8} - \frac{3}{91} a^{7} + \frac{194}{455} a^{6} - \frac{59}{455} a^{5} + \frac{1}{5} a^{4} - \frac{40}{91} a^{3} - \frac{1}{91} a^{2} + \frac{149}{455} a$, $\frac{1}{1365} a^{26} + \frac{1}{1365} a^{25} - \frac{17}{1365} a^{24} - \frac{8}{1365} a^{23} - \frac{8}{1365} a^{22} + \frac{73}{1365} a^{21} - \frac{41}{1365} a^{20} - \frac{113}{1365} a^{19} - \frac{332}{1365} a^{18} + \frac{46}{105} a^{17} + \frac{478}{1365} a^{16} - \frac{44}{195} a^{15} + \frac{142}{455} a^{14} + \frac{193}{455} a^{13} + \frac{16}{35} a^{12} + \frac{77}{195} a^{11} + \frac{86}{1365} a^{10} - \frac{187}{1365} a^{9} - \frac{248}{1365} a^{8} + \frac{22}{1365} a^{7} - \frac{94}{273} a^{6} - \frac{394}{1365} a^{5} + \frac{29}{195} a^{4} - \frac{643}{1365} a^{3} + \frac{577}{1365} a^{2} - \frac{446}{1365} a + \frac{8}{21}$, $\frac{1}{1365} a^{27} + \frac{1}{91} a^{22} - \frac{1}{455} a^{21} + \frac{1}{455} a^{20} - \frac{31}{91} a^{19} - \frac{36}{91} a^{18} - \frac{218}{455} a^{17} + \frac{23}{65} a^{16} - \frac{94}{195} a^{15} - \frac{20}{91} a^{14} - \frac{6}{91} a^{13} - \frac{46}{195} a^{12} + \frac{11}{35} a^{11} - \frac{223}{455} a^{10} + \frac{73}{273} a^{9} + \frac{37}{91} a^{8} + \frac{12}{65} a^{7} + \frac{487}{1365} a^{6} + \frac{28}{65} a^{5} + \frac{45}{91} a^{4} + \frac{10}{39} a^{3} - \frac{33}{455} a^{2} - \frac{71}{455} a - \frac{34}{105}$, $\frac{1}{2895165} a^{28} + \frac{698}{2895165} a^{27} - \frac{76}{965055} a^{26} + \frac{19}{27573} a^{25} + \frac{3359}{965055} a^{24} - \frac{122}{9555} a^{23} + \frac{992}{137865} a^{22} + \frac{65879}{965055} a^{21} + \frac{2437}{965055} a^{20} + \frac{36958}{137865} a^{19} - \frac{451886}{965055} a^{18} - \frac{51838}{965055} a^{17} + \frac{125659}{579033} a^{16} - \frac{926936}{2895165} a^{15} - \frac{18419}{74235} a^{14} + \frac{6014}{59085} a^{13} + \frac{1263523}{2895165} a^{12} - \frac{10775}{193011} a^{11} - \frac{170089}{413595} a^{10} - \frac{671618}{2895165} a^{9} + \frac{62289}{321685} a^{8} + \frac{252643}{2895165} a^{7} + \frac{297056}{2895165} a^{6} + \frac{312967}{965055} a^{5} + \frac{743807}{2895165} a^{4} - \frac{7933}{82719} a^{3} + \frac{49487}{193011} a^{2} + \frac{330989}{2895165} a + \frac{35167}{222705}$, $\frac{1}{5357245755316713577322521208268101268509257058530965} a^{29} + \frac{2105681886687898651378212906595729214975651}{5357245755316713577322521208268101268509257058530965} a^{28} - \frac{229592283243752748250861081056513806399198858864}{1785748585105571192440840402756033756169752352843655} a^{27} - \frac{306187709186331101962840632946725155511477029668}{1785748585105571192440840402756033756169752352843655} a^{26} - \frac{996203141233503754575140343102782705949634017451}{1785748585105571192440840402756033756169752352843655} a^{25} + \frac{930826051922416697526907985357028033854517900932}{1785748585105571192440840402756033756169752352843655} a^{24} + \frac{3221086043201968632090522204811397544192992881799}{1785748585105571192440840402756033756169752352843655} a^{23} + \frac{226591418904384456903661466184975531424130442167}{1785748585105571192440840402756033756169752352843655} a^{22} - \frac{157061752685692252553348606728773502193443976111021}{1785748585105571192440840402756033756169752352843655} a^{21} - \frac{66160186897758373132882452672989843852604132853106}{1785748585105571192440840402756033756169752352843655} a^{20} - \frac{94309256791116194743744798696182022783350606160405}{357149717021114238488168080551206751233950470568731} a^{19} - \frac{518159941646400684958408341547894309785443348847577}{1785748585105571192440840402756033756169752352843655} a^{18} + \frac{1607745010772756778405716730440480891949670320699917}{5357245755316713577322521208268101268509257058530965} a^{17} - \frac{43694744590697356257054341269825690445423190275543}{765320822188101939617503029752585895501322436932995} a^{16} - \frac{43538172828529106596765350995023152895774109219810}{357149717021114238488168080551206751233950470568731} a^{15} + \frac{1444603479616898565806776252634566567821593915395}{82419165466410978112654172434893865669373185515861} a^{14} - \frac{524383331471205098687607642542071787409047447123783}{5357245755316713577322521208268101268509257058530965} a^{13} - \frac{362549004632210487751893347334980899900753451232}{1735421365505900089835607777216748062361275367195} a^{12} - \frac{191617675516047485108745902424462726703868354370683}{487022341392428507029320109842554660773568823502815} a^{11} + \frac{479456967297769497244541973782921692839584047917113}{1071449151063342715464504241653620253701851411706193} a^{10} + \frac{254824415662408174040209206132886026292576403458606}{1785748585105571192440840402756033756169752352843655} a^{9} + \frac{1474184922282299846495737689750778781307317578113244}{5357245755316713577322521208268101268509257058530965} a^{8} + \frac{1670412942919418106007286425774311809241898107927782}{5357245755316713577322521208268101268509257058530965} a^{7} + \frac{63169215149278891026907600558596676695484675755476}{595249528368523730813613467585344585389917450947885} a^{6} - \frac{2398725740873179319410273767613074773065147975340794}{5357245755316713577322521208268101268509257058530965} a^{5} - \frac{206317619566771272568175591702033452230546214280713}{487022341392428507029320109842554660773568823502815} a^{4} + \frac{676914548269389650681249653920475431751827865927106}{1785748585105571192440840402756033756169752352843655} a^{3} + \frac{2166348437777484420295040896438121218975213560446954}{5357245755316713577322521208268101268509257058530965} a^{2} + \frac{213455260032699785502243280095747845885659520317568}{487022341392428507029320109842554660773568823502815} a - \frac{61574902093047757720674366751790998654116516363569}{137365275777351630187756954058156442782288642526435}$

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:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{28462002722786499284753032816814653700}{30317591421286120523162940091320603597} a^{29} + \frac{3619278397947896576200338827762898925}{10105863807095373507720980030440201199} a^{28} - \frac{1312298602345125295358981719662366075}{10105863807095373507720980030440201199} a^{27} + \frac{95405291063157126503097438468412958850}{10105863807095373507720980030440201199} a^{26} - \frac{36367531137290110774953308414864845037}{10105863807095373507720980030440201199} a^{25} + \frac{525527446311583792284086570769780616900}{10105863807095373507720980030440201199} a^{24} - \frac{1054635870958817756398204122562794807675}{10105863807095373507720980030440201199} a^{23} - \frac{2590143704661338513635887425110164341550}{10105863807095373507720980030440201199} a^{22} + \frac{850048619271101942131553191269164084275}{10105863807095373507720980030440201199} a^{21} + \frac{1975347934877957111038159693227220526040}{10105863807095373507720980030440201199} a^{20} - \frac{5523059093159433402198589864005018884750}{10105863807095373507720980030440201199} a^{19} + \frac{47070729815537056389278468858176099100750}{10105863807095373507720980030440201199} a^{18} - \frac{292134514893133712754268808069785634378575}{30317591421286120523162940091320603597} a^{17} + \frac{175165268244565669536665949744854008124725}{10105863807095373507720980030440201199} a^{16} - \frac{304896124545251530841200580017870270208315}{10105863807095373507720980030440201199} a^{15} + \frac{1319167661237719113450559475594377019183525}{30317591421286120523162940091320603597} a^{14} - \frac{654608145651314508466409569523125056233650}{10105863807095373507720980030440201199} a^{13} + \frac{64102155808919135990454711106791809841050}{777374139007336423670844617726169323} a^{12} - \frac{20582977701429532501585305462275082669800}{212011128820182661001139441198046179} a^{11} + \frac{1014306039119988302979095465236704918406070}{10105863807095373507720980030440201199} a^{10} - \frac{812612296543988678078742512747163667200500}{10105863807095373507720980030440201199} a^{9} + \frac{1569753066023245948014259975297822047518950}{30317591421286120523162940091320603597} a^{8} - \frac{166634965900769550112161104050885224034500}{10105863807095373507720980030440201199} a^{7} - \frac{92691151981952739613339147678644131865575}{10105863807095373507720980030440201199} a^{6} + \frac{410366956472347224214019824069103099378045}{30317591421286120523162940091320603597} a^{5} - \frac{8948261474047600771643735627409097112600}{918714891554124864338270911858200109} a^{4} + \frac{15419509118447175639631781418175953334125}{10105863807095373507720980030440201199} a^{3} + \frac{7607553036377624804569245063026768537150}{2332122417022009271012533853178507969} a^{2} - \frac{58803895853992073636283501160661010350}{70670376273394220333713147066015393} a - \frac{332626935564749037683632574285524203870}{777374139007336423670844617726169323} \) (order $6$)
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:  \( 17149795938.655872 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 17149795938.655872 \cdot 1}{6\sqrt{18201370075621419164235703647136688232421875}}\approx 0.629149262500347$ (assuming GRH)

Galois group

$D_{30}$ (as 30T14):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.135.1, 5.1.140625.1, 6.0.54675.1, 10.0.59326171875.1, 15.1.2463153133392333984375.1

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

Sibling fields

Degree 30 sibling: Deg 30

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $30$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $30$ $15^{2}$ $30$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ $30$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{15}$

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
$3$3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.11$x^{12} + 6 x^{11} + 21 x^{10} + 36 x^{9} + 30 x^{8} + 36 x^{7} + 3 x^{6} + 36 x^{5} + 27 x^{4} - 9 x^{2} + 36$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
3.12.14.11$x^{12} + 6 x^{11} + 21 x^{10} + 36 x^{9} + 30 x^{8} + 36 x^{7} + 3 x^{6} + 36 x^{5} + 27 x^{4} - 9 x^{2} + 36$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
5Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.5.2t1.a.a$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.15.2t1.a.a$1$ $ 3 \cdot 5 $ $x^{2} - x + 4$ $C_2$ (as 2T1) $1$ $-1$
* 1.3.2t1.a.a$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
* 2.135.6t3.a.a$2$ $ 3^{3} \cdot 5 $ $x^{6} - x^{3} - 1$ $D_{6}$ (as 6T3) $1$ $0$
* 2.135.3t2.b.a$2$ $ 3^{3} \cdot 5 $ $x^{3} + 3 x - 1$ $S_3$ (as 3T2) $1$ $0$
* 2.375.5t2.a.a$2$ $ 3 \cdot 5^{3}$ $x^{5} - 5 x^{2} - 3$ $D_{5}$ (as 5T2) $1$ $0$
* 2.375.5t2.a.b$2$ $ 3 \cdot 5^{3}$ $x^{5} - 5 x^{2} - 3$ $D_{5}$ (as 5T2) $1$ $0$
* 2.375.10t3.a.b$2$ $ 3 \cdot 5^{3}$ $x^{10} - 5 x^{7} + 10 x^{6} + 9 x^{5} - 10 x^{4} - 5 x^{3} - 1$ $D_{10}$ (as 10T3) $1$ $0$
* 2.375.10t3.a.a$2$ $ 3 \cdot 5^{3}$ $x^{10} - 5 x^{7} + 10 x^{6} + 9 x^{5} - 10 x^{4} - 5 x^{3} - 1$ $D_{10}$ (as 10T3) $1$ $0$
* 2.3375.15t2.a.a$2$ $ 3^{3} \cdot 5^{3}$ $x^{15} - 10 x^{12} + 15 x^{11} - 24 x^{10} + 95 x^{9} - 90 x^{8} + 90 x^{7} - 125 x^{6} + 27 x^{5} + 90 x^{4} + 120 x^{3} + 195 x^{2} + 90 x + 33$ $D_{15}$ (as 15T2) $1$ $0$
* 2.3375.30t14.a.c$2$ $ 3^{3} \cdot 5^{3}$ $x^{30} - 10 x^{27} - 54 x^{25} + 90 x^{24} + 315 x^{23} + 15 x^{22} - 240 x^{21} + 504 x^{20} - 4740 x^{19} + 8375 x^{18} - 14580 x^{17} + 25140 x^{16} - 34179 x^{15} + 51465 x^{14} - 61710 x^{13} + 70215 x^{12} - 67800 x^{11} + 45276 x^{10} - 22840 x^{9} - 3240 x^{8} + 16350 x^{7} - 10730 x^{6} + 4956 x^{5} + 2280 x^{4} - 4065 x^{3} - 420 x^{2} + 780 x + 169$ $D_{30}$ (as 30T14) $1$ $0$
* 2.3375.15t2.a.c$2$ $ 3^{3} \cdot 5^{3}$ $x^{15} - 10 x^{12} + 15 x^{11} - 24 x^{10} + 95 x^{9} - 90 x^{8} + 90 x^{7} - 125 x^{6} + 27 x^{5} + 90 x^{4} + 120 x^{3} + 195 x^{2} + 90 x + 33$ $D_{15}$ (as 15T2) $1$ $0$
* 2.3375.30t14.a.a$2$ $ 3^{3} \cdot 5^{3}$ $x^{30} - 10 x^{27} - 54 x^{25} + 90 x^{24} + 315 x^{23} + 15 x^{22} - 240 x^{21} + 504 x^{20} - 4740 x^{19} + 8375 x^{18} - 14580 x^{17} + 25140 x^{16} - 34179 x^{15} + 51465 x^{14} - 61710 x^{13} + 70215 x^{12} - 67800 x^{11} + 45276 x^{10} - 22840 x^{9} - 3240 x^{8} + 16350 x^{7} - 10730 x^{6} + 4956 x^{5} + 2280 x^{4} - 4065 x^{3} - 420 x^{2} + 780 x + 169$ $D_{30}$ (as 30T14) $1$ $0$
* 2.3375.15t2.a.b$2$ $ 3^{3} \cdot 5^{3}$ $x^{15} - 10 x^{12} + 15 x^{11} - 24 x^{10} + 95 x^{9} - 90 x^{8} + 90 x^{7} - 125 x^{6} + 27 x^{5} + 90 x^{4} + 120 x^{3} + 195 x^{2} + 90 x + 33$ $D_{15}$ (as 15T2) $1$ $0$
* 2.3375.30t14.a.b$2$ $ 3^{3} \cdot 5^{3}$ $x^{30} - 10 x^{27} - 54 x^{25} + 90 x^{24} + 315 x^{23} + 15 x^{22} - 240 x^{21} + 504 x^{20} - 4740 x^{19} + 8375 x^{18} - 14580 x^{17} + 25140 x^{16} - 34179 x^{15} + 51465 x^{14} - 61710 x^{13} + 70215 x^{12} - 67800 x^{11} + 45276 x^{10} - 22840 x^{9} - 3240 x^{8} + 16350 x^{7} - 10730 x^{6} + 4956 x^{5} + 2280 x^{4} - 4065 x^{3} - 420 x^{2} + 780 x + 169$ $D_{30}$ (as 30T14) $1$ $0$
* 2.3375.15t2.a.d$2$ $ 3^{3} \cdot 5^{3}$ $x^{15} - 10 x^{12} + 15 x^{11} - 24 x^{10} + 95 x^{9} - 90 x^{8} + 90 x^{7} - 125 x^{6} + 27 x^{5} + 90 x^{4} + 120 x^{3} + 195 x^{2} + 90 x + 33$ $D_{15}$ (as 15T2) $1$ $0$
* 2.3375.30t14.a.d$2$ $ 3^{3} \cdot 5^{3}$ $x^{30} - 10 x^{27} - 54 x^{25} + 90 x^{24} + 315 x^{23} + 15 x^{22} - 240 x^{21} + 504 x^{20} - 4740 x^{19} + 8375 x^{18} - 14580 x^{17} + 25140 x^{16} - 34179 x^{15} + 51465 x^{14} - 61710 x^{13} + 70215 x^{12} - 67800 x^{11} + 45276 x^{10} - 22840 x^{9} - 3240 x^{8} + 16350 x^{7} - 10730 x^{6} + 4956 x^{5} + 2280 x^{4} - 4065 x^{3} - 420 x^{2} + 780 x + 169$ $D_{30}$ (as 30T14) $1$ $0$

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 *.