Properties

Label 28.2.265...875.1
Degree $28$
Signature $[2, 13]$
Discriminant $-2.657\times 10^{45}$
Root discriminant $41.91$
Ramified primes $3, 5, 71$
Class number not computed
Class group not computed
Galois group $D_{28}$ (as 28T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 9*x^27 + 53*x^26 - 226*x^25 + 695*x^24 - 1455*x^23 + 1352*x^22 + 4587*x^21 - 26437*x^20 + 75511*x^19 - 148243*x^18 + 166438*x^17 - 8850*x^16 - 553583*x^15 + 1599191*x^14 - 2514414*x^13 + 2904360*x^12 - 819447*x^11 - 1193138*x^10 + 5517125*x^9 - 17624126*x^8 + 7384001*x^7 - 17552991*x^6 + 11860646*x^5 - 8713168*x^4 - 7406098*x^3 - 9688274*x^2 - 1853761*x - 465059)
 
gp: K = bnfinit(x^28 - 9*x^27 + 53*x^26 - 226*x^25 + 695*x^24 - 1455*x^23 + 1352*x^22 + 4587*x^21 - 26437*x^20 + 75511*x^19 - 148243*x^18 + 166438*x^17 - 8850*x^16 - 553583*x^15 + 1599191*x^14 - 2514414*x^13 + 2904360*x^12 - 819447*x^11 - 1193138*x^10 + 5517125*x^9 - 17624126*x^8 + 7384001*x^7 - 17552991*x^6 + 11860646*x^5 - 8713168*x^4 - 7406098*x^3 - 9688274*x^2 - 1853761*x - 465059, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-465059, -1853761, -9688274, -7406098, -8713168, 11860646, -17552991, 7384001, -17624126, 5517125, -1193138, -819447, 2904360, -2514414, 1599191, -553583, -8850, 166438, -148243, 75511, -26437, 4587, 1352, -1455, 695, -226, 53, -9, 1]);
 

\( x^{28} - 9 x^{27} + 53 x^{26} - 226 x^{25} + 695 x^{24} - 1455 x^{23} + 1352 x^{22} + 4587 x^{21} - 26437 x^{20} + 75511 x^{19} - 148243 x^{18} + 166438 x^{17} - 8850 x^{16} - 553583 x^{15} + 1599191 x^{14} - 2514414 x^{13} + 2904360 x^{12} - 819447 x^{11} - 1193138 x^{10} + 5517125 x^{9} - 17624126 x^{8} + 7384001 x^{7} - 17552991 x^{6} + 11860646 x^{5} - 8713168 x^{4} - 7406098 x^{3} - 9688274 x^{2} - 1853761 x - 465059 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-2657211910834657108824251865653514862060546875\)\(\medspace = -\,3^{14}\cdot 5^{21}\cdot 71^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $41.91$
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:  $3, 5, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{7} a^{18} - \frac{1}{7} a^{17} - \frac{3}{7} a^{14} + \frac{1}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{10} + \frac{2}{7} a^{9} + \frac{2}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{19} - \frac{1}{7} a^{17} - \frac{3}{7} a^{15} - \frac{2}{7} a^{14} + \frac{2}{7} a^{13} + \frac{1}{7} a^{12} + \frac{2}{7} a^{11} - \frac{3}{7} a^{10} + \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{20} - \frac{1}{7} a^{17} - \frac{3}{7} a^{16} - \frac{2}{7} a^{15} - \frac{1}{7} a^{14} + \frac{2}{7} a^{13} + \frac{3}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{9} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{21} + \frac{3}{7} a^{17} - \frac{2}{7} a^{16} - \frac{1}{7} a^{15} - \frac{1}{7} a^{14} - \frac{3}{7} a^{13} - \frac{2}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{2}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{7} a^{22} + \frac{1}{7} a^{17} - \frac{1}{7} a^{16} - \frac{1}{7} a^{15} - \frac{1}{7} a^{14} + \frac{2}{7} a^{13} + \frac{1}{7} a^{12} - \frac{1}{7} a^{11} + \frac{3}{7} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{91} a^{23} + \frac{2}{91} a^{21} - \frac{4}{91} a^{20} + \frac{4}{91} a^{19} - \frac{6}{91} a^{18} - \frac{9}{91} a^{17} - \frac{5}{13} a^{16} + \frac{5}{13} a^{15} - \frac{32}{91} a^{14} + \frac{2}{91} a^{13} - \frac{34}{91} a^{12} + \frac{17}{91} a^{11} + \frac{1}{7} a^{10} + \frac{22}{91} a^{9} + \frac{27}{91} a^{8} - \frac{18}{91} a^{7} + \frac{25}{91} a^{6} - \frac{27}{91} a^{5} - \frac{3}{7} a^{4} + \frac{20}{91} a^{3} - \frac{4}{91} a^{2} - \frac{5}{91} a - \frac{4}{13}$, $\frac{1}{71071} a^{24} - \frac{290}{71071} a^{23} - \frac{334}{10153} a^{22} - \frac{584}{71071} a^{21} + \frac{1944}{71071} a^{20} - \frac{106}{6461} a^{19} + \frac{2004}{71071} a^{18} + \frac{32397}{71071} a^{17} + \frac{4523}{10153} a^{16} - \frac{27875}{71071} a^{15} - \frac{1217}{5467} a^{14} + \frac{18561}{71071} a^{13} - \frac{4491}{10153} a^{12} - \frac{33673}{71071} a^{11} - \frac{3098}{71071} a^{10} + \frac{32920}{71071} a^{9} + \frac{12484}{71071} a^{8} - \frac{18012}{71071} a^{7} + \frac{11716}{71071} a^{6} - \frac{4152}{10153} a^{5} - \frac{17660}{71071} a^{4} + \frac{3123}{6461} a^{3} - \frac{1390}{6461} a^{2} - \frac{29908}{71071} a - \frac{4157}{10153}$, $\frac{1}{71071} a^{25} + \frac{23}{6461} a^{23} + \frac{1647}{71071} a^{22} - \frac{4187}{71071} a^{21} + \frac{2617}{71071} a^{20} + \frac{475}{71071} a^{19} + \frac{2034}{71071} a^{18} + \frac{1502}{6461} a^{17} - \frac{166}{923} a^{16} - \frac{19345}{71071} a^{15} + \frac{27437}{71071} a^{14} - \frac{18903}{71071} a^{13} + \frac{14632}{71071} a^{12} - \frac{9673}{71071} a^{11} + \frac{27964}{71071} a^{10} - \frac{16557}{71071} a^{9} + \frac{13653}{71071} a^{8} + \frac{20155}{71071} a^{7} + \frac{12619}{71071} a^{6} - \frac{34850}{71071} a^{5} - \frac{394}{71071} a^{4} + \frac{450}{6461} a^{3} + \frac{250}{10153} a^{2} + \frac{1089}{6461} a + \frac{1115}{10153}$, $\frac{1}{600478879} a^{26} + \frac{809}{600478879} a^{25} + \frac{729}{600478879} a^{24} + \frac{869590}{600478879} a^{23} + \frac{4438996}{600478879} a^{22} - \frac{957137}{85782697} a^{21} - \frac{32895524}{600478879} a^{20} - \frac{3422414}{54588989} a^{19} + \frac{284860}{4199153} a^{18} + \frac{154790400}{600478879} a^{17} + \frac{8899335}{85782697} a^{16} - \frac{14399713}{600478879} a^{15} + \frac{20378690}{46190683} a^{14} + \frac{198118223}{600478879} a^{13} + \frac{12620827}{54588989} a^{12} + \frac{297909452}{600478879} a^{11} - \frac{10554207}{54588989} a^{10} - \frac{144489837}{600478879} a^{9} + \frac{108597248}{600478879} a^{8} - \frac{71089838}{600478879} a^{7} + \frac{85248803}{600478879} a^{6} - \frac{150015035}{600478879} a^{5} - \frac{13562937}{600478879} a^{4} - \frac{97919436}{600478879} a^{3} - \frac{282181388}{600478879} a^{2} + \frac{3041856}{12254671} a - \frac{5485143}{12254671}$, $\frac{1}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{27} + \frac{5373516622068571055929704525672100813052722904985247559325109034184826223866230361}{10076418005841588225514215404121412397980476238418168302744297014465193518899319125114737961} a^{26} - \frac{41675264298300723126351995801154911165215538018039228165635457533827730088301061568109}{6412266003717374325327227984440898798714848515357016192655461736477850421117748534163924157} a^{25} - \frac{308389066239012866985533396861931141137039437944769993408938449136351160824357790427709}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{24} - \frac{364818860050851310258263202939790958103376111472159216105470445354351112374313129881890309}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{23} + \frac{3300934317694941333913983097591828013229571626532499290880512398727494484119550647437}{276274939742706300957668641395848475701266842150537110691089860839132317689273047256061} a^{22} - \frac{288370274361647183293331521770737030373030479663082805955498815657354218926159008245557103}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{21} - \frac{3441801167736393350398766049085469273840873923318041334054544319842514375601917965575159955}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{20} + \frac{938786291353049276244517432517107644155269133327906112798552289031546608292549815478858060}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{19} - \frac{68206560617633500298716144198769426385679363866834994748976554821022285920857960145351540}{5425763541607009044507654448373068214297179512994398316862313777019719587099633375061781979} a^{18} - \frac{3941815836030607801280433910879583816261018385628394912799278144174399083736874199497558526}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{17} - \frac{19858306208923117779142996380301153568554220302922495041752107483156450387594406268869448911}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{16} - \frac{5384611587022614279034899425262731786669926281009007670855186581026062137720249372787430}{67626966482158310238350438953834982536781719720927304045263738352115392744290732383320389} a^{15} + \frac{23926127355456043008374986057650657161939001509891182228758520281156374230022363567954202}{110729868196061409071584784660674861516268969652946904425761505653463665042849660715546571} a^{14} - \frac{1915866363005282480757144143279135794045055920223022345051867662077754315120495309135946644}{10076418005841588225514215404121412397980476238418168302744297014465193518899319125114737961} a^{13} - \frac{2353330059833690361820610082990045479707994373068532077243140935325599620732146588725567327}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{12} + \frac{163321643684216905704735043282882350764793998306358044446787238762119015139136317252091038}{775109077372429863501093492624724030613882787570628330980330539574245655299947625008825997} a^{11} + \frac{22297954384159468271900868040586871273755768537347848138733520047564777473947942756920392495}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{10} - \frac{11986192321875868595470260661629070327254094139541121445092954701458631022578139558985712}{24230479574335663888216938450309133213968853888329501243287557231623618904945116412161857} a^{9} + \frac{13447654363053856525136966257031034128422099152769191517613998886538412685632320636963950992}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{8} - \frac{1017304675347541427369219533602392758504243251922854492594892168874315350631199457315015198}{10076418005841588225514215404121412397980476238418168302744297014465193518899319125114737961} a^{7} - \frac{2823724326636737523779776254814318497717064707413271225224097841871969456413257669998305994}{10076418005841588225514215404121412397980476238418168302744297014465193518899319125114737961} a^{6} - \frac{2939971536585818985352564773023687308110662184416593352242612342711314841794690841953915322}{6412266003717374325327227984440898798714848515357016192655461736477850421117748534163924157} a^{5} + \frac{3905735719275329432825239965021075541576497439002075670839398749643079794139814976981177891}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{4} - \frac{17691472126040151849560752464878030700807447365816106472900771802666544646489613046720439044}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{3} - \frac{12064724067120814243174164766066458687634638168107005420689340684129121983929463120841247910}{70534926040891117578599507828849886785863333668927178119210079101256354632295233875803165727} a^{2} + \frac{78606142670275890289344875638738522726005125733788402679302932514179128973396177757174207}{592730470931858130912600906124788964587086837554009900161429236145011383464665830889102233} a - \frac{254798805366025695765878704850954490551732481011170407777308503175365953558741626288271217}{1439488286548798317930602200588773199711496605488309757534899573495027645557045589302105423}$

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

Class group and class number

not computed

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:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$D_{28}$ (as 28T10):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 56
The 17 conjugacy class representatives for $D_{28}$
Character table for $D_{28}$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.79875.1, 7.1.357911.1, 14.2.10007834681328125.1

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

Sibling fields

Degree 28 sibling: Deg 28

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $28$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $28$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $28$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.355.2t1.a.a$1$ $ 5 \cdot 71 $ \(\Q(\sqrt{-355}) \) $C_2$ (as 2T1) $1$ $-1$
1.71.2t1.a.a$1$ $ 71 $ \(\Q(\sqrt{-71}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.15975.4t3.a.a$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 4.0.5671125.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.1775.14t3.a.c$2$ $ 5^{2} \cdot 71 $ 14.0.710556262374296875.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.1775.14t3.a.b$2$ $ 5^{2} \cdot 71 $ 14.0.710556262374296875.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.c$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.71.7t2.a.b$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.1775.14t3.a.a$2$ $ 5^{2} \cdot 71 $ 14.0.710556262374296875.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.a$2$ $ 71 $ 7.1.357911.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.15975.28t10.a.d$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 28.2.2657211910834657108824251865653514862060546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.15975.28t10.a.c$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 28.2.2657211910834657108824251865653514862060546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.15975.28t10.a.e$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 28.2.2657211910834657108824251865653514862060546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.15975.28t10.a.a$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 28.2.2657211910834657108824251865653514862060546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.15975.28t10.a.b$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 28.2.2657211910834657108824251865653514862060546875.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.15975.28t10.a.f$2$ $ 3^{2} \cdot 5^{2} \cdot 71 $ 28.2.2657211910834657108824251865653514862060546875.1 $D_{28}$ (as 28T10) $1$ $0$

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 *.