Properties

Label 28.2.462...167.1
Degree $28$
Signature $[2, 13]$
Discriminant $-4.620\times 10^{43}$
Root discriminant $36.26$
Ramified primes $7, 71$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{28}$ (as 28T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 3*x^27 + 5*x^26 - 37*x^25 + 223*x^24 - 471*x^23 + 612*x^22 - 2497*x^21 + 8709*x^20 - 13185*x^19 + 9400*x^18 - 14827*x^17 + 46063*x^16 - 65223*x^15 + 33754*x^14 - 24986*x^13 + 51170*x^12 - 43382*x^11 - 84023*x^10 + 134581*x^9 - 102493*x^8 + 27983*x^7 - 169214*x^6 + 162210*x^5 - 64809*x^4 - 131662*x^3 + 54254*x^2 - 97483*x - 91661)
 
gp: K = bnfinit(x^28 - 3*x^27 + 5*x^26 - 37*x^25 + 223*x^24 - 471*x^23 + 612*x^22 - 2497*x^21 + 8709*x^20 - 13185*x^19 + 9400*x^18 - 14827*x^17 + 46063*x^16 - 65223*x^15 + 33754*x^14 - 24986*x^13 + 51170*x^12 - 43382*x^11 - 84023*x^10 + 134581*x^9 - 102493*x^8 + 27983*x^7 - 169214*x^6 + 162210*x^5 - 64809*x^4 - 131662*x^3 + 54254*x^2 - 97483*x - 91661, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-91661, -97483, 54254, -131662, -64809, 162210, -169214, 27983, -102493, 134581, -84023, -43382, 51170, -24986, 33754, -65223, 46063, -14827, 9400, -13185, 8709, -2497, 612, -471, 223, -37, 5, -3, 1]);
 

\( x^{28} - 3 x^{27} + 5 x^{26} - 37 x^{25} + 223 x^{24} - 471 x^{23} + 612 x^{22} - 2497 x^{21} + 8709 x^{20} - 13185 x^{19} + 9400 x^{18} - 14827 x^{17} + 46063 x^{16} - 65223 x^{15} + 33754 x^{14} - 24986 x^{13} + 51170 x^{12} - 43382 x^{11} - 84023 x^{10} + 134581 x^{9} - 102493 x^{8} + 27983 x^{7} - 169214 x^{6} + 162210 x^{5} - 64809 x^{4} - 131662 x^{3} + 54254 x^{2} - 97483 x - 91661 \)

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:  \(-46203590105972724234309704622025792939166167\)\(\medspace = -\,7^{21}\cdot 71^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $36.26$
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:  $7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{11} a^{22} + \frac{1}{11} a^{21} + \frac{4}{11} a^{20} - \frac{3}{11} a^{17} - \frac{5}{11} a^{16} - \frac{5}{11} a^{15} - \frac{5}{11} a^{14} + \frac{5}{11} a^{13} + \frac{2}{11} a^{12} + \frac{2}{11} a^{11} - \frac{1}{11} a^{10} - \frac{4}{11} a^{9} + \frac{4}{11} a^{8} + \frac{5}{11} a^{7} - \frac{3}{11} a^{6} + \frac{4}{11} a^{5} - \frac{1}{11} a^{4} + \frac{5}{11} a^{3} - \frac{4}{11} a^{2} - \frac{4}{11} a + \frac{5}{11}$, $\frac{1}{11} a^{23} + \frac{3}{11} a^{21} - \frac{4}{11} a^{20} - \frac{3}{11} a^{18} - \frac{2}{11} a^{17} - \frac{1}{11} a^{14} - \frac{3}{11} a^{13} - \frac{3}{11} a^{11} - \frac{3}{11} a^{10} - \frac{3}{11} a^{9} + \frac{1}{11} a^{8} + \frac{3}{11} a^{7} - \frac{4}{11} a^{6} - \frac{5}{11} a^{5} - \frac{5}{11} a^{4} + \frac{2}{11} a^{3} - \frac{2}{11} a - \frac{5}{11}$, $\frac{1}{187} a^{24} + \frac{7}{187} a^{23} + \frac{1}{187} a^{22} - \frac{40}{187} a^{21} + \frac{30}{187} a^{20} - \frac{36}{187} a^{19} + \frac{10}{187} a^{18} + \frac{91}{187} a^{17} + \frac{43}{187} a^{16} + \frac{42}{187} a^{15} + \frac{4}{17} a^{14} + \frac{13}{187} a^{13} - \frac{40}{187} a^{12} - \frac{1}{11} a^{11} - \frac{2}{17} a^{10} - \frac{56}{187} a^{9} + \frac{90}{187} a^{8} + \frac{73}{187} a^{7} - \frac{82}{187} a^{6} - \frac{15}{187} a^{5} - \frac{75}{187} a^{4} - \frac{29}{187} a^{3} - \frac{71}{187} a^{2} - \frac{7}{17} a + \frac{32}{187}$, $\frac{1}{12529} a^{25} + \frac{1}{737} a^{24} - \frac{490}{12529} a^{23} + \frac{89}{12529} a^{22} + \frac{3676}{12529} a^{21} + \frac{1862}{12529} a^{20} - \frac{1846}{12529} a^{19} - \frac{3549}{12529} a^{18} - \frac{3518}{12529} a^{17} - \frac{497}{12529} a^{16} + \frac{2861}{12529} a^{15} - \frac{2199}{12529} a^{14} + \frac{2742}{12529} a^{13} - \frac{5415}{12529} a^{12} - \frac{3694}{12529} a^{11} + \frac{2597}{12529} a^{10} - \frac{120}{1139} a^{9} + \frac{3132}{12529} a^{8} - \frac{278}{1139} a^{7} - \frac{15}{187} a^{6} + \frac{5487}{12529} a^{5} + \frac{598}{12529} a^{4} - \frac{1262}{12529} a^{3} - \frac{5190}{12529} a^{2} + \frac{2900}{12529} a + \frac{541}{12529}$, $\frac{1}{5049187} a^{26} - \frac{80}{5049187} a^{25} + \frac{4427}{5049187} a^{24} - \frac{207115}{5049187} a^{23} + \frac{91590}{5049187} a^{22} + \frac{2442004}{5049187} a^{21} - \frac{477528}{5049187} a^{20} + \frac{16108}{459017} a^{19} + \frac{62003}{459017} a^{18} + \frac{417732}{5049187} a^{17} - \frac{1306752}{5049187} a^{16} - \frac{733}{2077} a^{15} - \frac{32583}{459017} a^{14} - \frac{37542}{297011} a^{13} - \frac{262741}{5049187} a^{12} + \frac{266378}{5049187} a^{11} - \frac{92068}{297011} a^{10} + \frac{468517}{5049187} a^{9} - \frac{43947}{388399} a^{8} + \frac{535749}{5049187} a^{7} + \frac{184176}{5049187} a^{6} - \frac{646077}{5049187} a^{5} + \frac{117760}{297011} a^{4} + \frac{44485}{162877} a^{3} + \frac{2223607}{5049187} a^{2} + \frac{2136333}{5049187} a + \frac{2474361}{5049187}$, $\frac{1}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{27} - \frac{3266431728120510730306151843161017174109308476054795686759813013225098086}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{26} - \frac{1074605403445941987643212923365545668201812253645247272514278451174293383752}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{25} + \frac{48299566080993007939057509139515998246806841887693564717020081725746678224854}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{24} + \frac{39718152249789771916423601930969701161144352986604745134045949859981031084410}{3123289383792564452623579530394340650217184081325082168770557010643069115179237} a^{23} - \frac{362972551171239712750760570835007591707209094939702638790658912118714239937462}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{22} - \frac{11846416717830468800888630105273509407713998632104793674557056946273649982214499}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{21} + \frac{7696378381614860179948849325781162978022915874869712043822282264884015733999312}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{20} + \frac{14014005907649812686352595436847863959022700985737258748310882643896341082766811}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{19} + \frac{5273384132488920455638291620344261523077705494929236465644808613159191638266505}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{18} - \frac{15332022137087646259250718691195573702129413036658237153923968977916567869492603}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{17} + \frac{2564980982790209616452529068278505820763715387308302376408074824185202434142001}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{16} - \frac{1436107900025928106481750694581192405179349618690052235476413109164178180823516}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{15} - \frac{5955535945217710079027965232693984320938829580791493607940905250146584949763015}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{14} - \frac{2583259643476960290457198574557744267880595105861789955420379642395875027176738}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{13} - \frac{12741794971656962092416509256603230950739855219726948074551371666849881404597718}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{12} + \frac{217923025169152645177866273630372363212640032665778910117985478356607901276021}{512778854055495656400886191557279808244612311859341848604121300254832242790621} a^{11} + \frac{11491813938761187743975346876705486056210311884085945947554548761280342560657615}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{10} - \frac{216758381047177359010951093804815096471935024516678165242236896351570843471910}{1108263974894135773511592736591540230722226609502448511499229907002379363450697} a^{9} + \frac{13928998681919094921204073117756009305486271199457697437035904588766901409948679}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{8} - \frac{400248977870112095586158447334969139239994771288783939547838679678688349460561}{1493747096596443868646059775405989006625609778025039298107657700742337402911809} a^{7} - \frac{14855871162958355818600009969155745188386124008915476102378163004387060961409}{512778854055495656400886191557279808244612311859341848604121300254832242790621} a^{6} + \frac{12604540398310573435346693577391652469014470274918132658379943887845642053739204}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{5} + \frac{14435102124320126836020439469191828907112629657079111675515213818083242648647928}{34356183221718208978859374834337747152389024894575903856476127117073760266971607} a^{4} + \frac{961901106240177960556656680356254063398253197074055696731185136074459655748900}{2020951954218718175227022049078691008964060287916229638616242771592574133351271} a^{3} - \frac{998558049441662501012986081976385578677446365730374988137475263216779552395529}{2020951954218718175227022049078691008964060287916229638616242771592574133351271} a^{2} - \frac{95701772299303229784393684442427472424612215057493133471878827524950090637817}{2642783324747554536835336525718288242491463453428915681267394393621058482074739} a + \frac{2196390047520776972436773670976436644869048710849995948876675029572035183201405}{34356183221718208978859374834337747152389024894575903856476127117073760266971607}$

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:  \( -1 \) (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:  \( 46802852168.54651 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{13}\cdot 46802852168.54651 \cdot 1}{2\sqrt{46203590105972724234309704622025792939166167}}\approx 0.327568960813624$ (assuming GRH)

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

\(\Q(\sqrt{497}) \), 4.2.1729063.1, 7.1.357911.1, 14.2.7490222540601799313.1

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

Sibling fields

Degree 28 sibling: 28.0.650754790224967946962108515803180182241777.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.7.0.1}{7} }^{4}$ $28$ $28$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $28$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$7$7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$71$71.2.1.2$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$$[\ ]_{2}$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.497.2t1.a.a$1$ $ 7 \cdot 71 $ $x^{2} - x - 124$ $C_2$ (as 2T1) $1$ $1$
1.71.2t1.a.a$1$ $ 71 $ $x^{2} - x + 18$ $C_2$ (as 2T1) $1$ $-1$
1.7.2t1.a.a$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
* 2.3479.4t3.c.a$2$ $ 7^{2} \cdot 71 $ $x^{4} + 7 x^{2} - 112$ $D_{4}$ (as 4T3) $1$ $0$
* 2.3479.14t3.a.a$2$ $ 7^{2} \cdot 71 $ $x^{14} - 4 x^{13} - 4 x^{12} + 32 x^{11} - 44 x^{10} + 76 x^{9} + 222 x^{8} - 1472 x^{7} + 483 x^{6} + 4460 x^{5} - 2122 x^{4} - 5110 x^{3} + 564 x^{2} + 2882 x - 1573$ $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.c$2$ $ 71 $ $x^{7} - x^{6} - x^{5} + x^{4} - x^{3} - x^{2} + 2 x + 1$ $D_{7}$ (as 7T2) $1$ $0$
* 2.3479.14t3.a.b$2$ $ 7^{2} \cdot 71 $ $x^{14} - 4 x^{13} - 4 x^{12} + 32 x^{11} - 44 x^{10} + 76 x^{9} + 222 x^{8} - 1472 x^{7} + 483 x^{6} + 4460 x^{5} - 2122 x^{4} - 5110 x^{3} + 564 x^{2} + 2882 x - 1573$ $D_{14}$ (as 14T3) $1$ $0$
* 2.71.7t2.a.a$2$ $ 71 $ $x^{7} - x^{6} - x^{5} + x^{4} - x^{3} - x^{2} + 2 x + 1$ $D_{7}$ (as 7T2) $1$ $0$
* 2.71.7t2.a.b$2$ $ 71 $ $x^{7} - x^{6} - x^{5} + x^{4} - x^{3} - x^{2} + 2 x + 1$ $D_{7}$ (as 7T2) $1$ $0$
* 2.3479.14t3.a.c$2$ $ 7^{2} \cdot 71 $ $x^{14} - 4 x^{13} - 4 x^{12} + 32 x^{11} - 44 x^{10} + 76 x^{9} + 222 x^{8} - 1472 x^{7} + 483 x^{6} + 4460 x^{5} - 2122 x^{4} - 5110 x^{3} + 564 x^{2} + 2882 x - 1573$ $D_{14}$ (as 14T3) $1$ $0$
* 2.3479.28t10.a.c$2$ $ 7^{2} \cdot 71 $ $x^{28} - 3 x^{27} + 5 x^{26} - 37 x^{25} + 223 x^{24} - 471 x^{23} + 612 x^{22} - 2497 x^{21} + 8709 x^{20} - 13185 x^{19} + 9400 x^{18} - 14827 x^{17} + 46063 x^{16} - 65223 x^{15} + 33754 x^{14} - 24986 x^{13} + 51170 x^{12} - 43382 x^{11} - 84023 x^{10} + 134581 x^{9} - 102493 x^{8} + 27983 x^{7} - 169214 x^{6} + 162210 x^{5} - 64809 x^{4} - 131662 x^{3} + 54254 x^{2} - 97483 x - 91661$ $D_{28}$ (as 28T10) $1$ $0$
* 2.3479.28t10.a.f$2$ $ 7^{2} \cdot 71 $ $x^{28} - 3 x^{27} + 5 x^{26} - 37 x^{25} + 223 x^{24} - 471 x^{23} + 612 x^{22} - 2497 x^{21} + 8709 x^{20} - 13185 x^{19} + 9400 x^{18} - 14827 x^{17} + 46063 x^{16} - 65223 x^{15} + 33754 x^{14} - 24986 x^{13} + 51170 x^{12} - 43382 x^{11} - 84023 x^{10} + 134581 x^{9} - 102493 x^{8} + 27983 x^{7} - 169214 x^{6} + 162210 x^{5} - 64809 x^{4} - 131662 x^{3} + 54254 x^{2} - 97483 x - 91661$ $D_{28}$ (as 28T10) $1$ $0$
* 2.3479.28t10.a.d$2$ $ 7^{2} \cdot 71 $ $x^{28} - 3 x^{27} + 5 x^{26} - 37 x^{25} + 223 x^{24} - 471 x^{23} + 612 x^{22} - 2497 x^{21} + 8709 x^{20} - 13185 x^{19} + 9400 x^{18} - 14827 x^{17} + 46063 x^{16} - 65223 x^{15} + 33754 x^{14} - 24986 x^{13} + 51170 x^{12} - 43382 x^{11} - 84023 x^{10} + 134581 x^{9} - 102493 x^{8} + 27983 x^{7} - 169214 x^{6} + 162210 x^{5} - 64809 x^{4} - 131662 x^{3} + 54254 x^{2} - 97483 x - 91661$ $D_{28}$ (as 28T10) $1$ $0$
* 2.3479.28t10.a.a$2$ $ 7^{2} \cdot 71 $ $x^{28} - 3 x^{27} + 5 x^{26} - 37 x^{25} + 223 x^{24} - 471 x^{23} + 612 x^{22} - 2497 x^{21} + 8709 x^{20} - 13185 x^{19} + 9400 x^{18} - 14827 x^{17} + 46063 x^{16} - 65223 x^{15} + 33754 x^{14} - 24986 x^{13} + 51170 x^{12} - 43382 x^{11} - 84023 x^{10} + 134581 x^{9} - 102493 x^{8} + 27983 x^{7} - 169214 x^{6} + 162210 x^{5} - 64809 x^{4} - 131662 x^{3} + 54254 x^{2} - 97483 x - 91661$ $D_{28}$ (as 28T10) $1$ $0$
* 2.3479.28t10.a.b$2$ $ 7^{2} \cdot 71 $ $x^{28} - 3 x^{27} + 5 x^{26} - 37 x^{25} + 223 x^{24} - 471 x^{23} + 612 x^{22} - 2497 x^{21} + 8709 x^{20} - 13185 x^{19} + 9400 x^{18} - 14827 x^{17} + 46063 x^{16} - 65223 x^{15} + 33754 x^{14} - 24986 x^{13} + 51170 x^{12} - 43382 x^{11} - 84023 x^{10} + 134581 x^{9} - 102493 x^{8} + 27983 x^{7} - 169214 x^{6} + 162210 x^{5} - 64809 x^{4} - 131662 x^{3} + 54254 x^{2} - 97483 x - 91661$ $D_{28}$ (as 28T10) $1$ $0$
* 2.3479.28t10.a.e$2$ $ 7^{2} \cdot 71 $ $x^{28} - 3 x^{27} + 5 x^{26} - 37 x^{25} + 223 x^{24} - 471 x^{23} + 612 x^{22} - 2497 x^{21} + 8709 x^{20} - 13185 x^{19} + 9400 x^{18} - 14827 x^{17} + 46063 x^{16} - 65223 x^{15} + 33754 x^{14} - 24986 x^{13} + 51170 x^{12} - 43382 x^{11} - 84023 x^{10} + 134581 x^{9} - 102493 x^{8} + 27983 x^{7} - 169214 x^{6} + 162210 x^{5} - 64809 x^{4} - 131662 x^{3} + 54254 x^{2} - 97483 x - 91661$ $D_{28}$ (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 *.