Properties

Label 28.0.192...877.1
Degree $28$
Signature $[0, 14]$
Discriminant $1.924\times 10^{45}$
Root discriminant $41.43$
Ramified primes $43, 53$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_{28}$ (as 28T10)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 12*x^27 + 74*x^26 - 288*x^25 + 773*x^24 - 1407*x^23 + 1601*x^22 - 676*x^21 - 794*x^20 + 2173*x^19 - 6644*x^18 + 23336*x^17 - 53512*x^16 + 82551*x^15 - 80880*x^14 + 23133*x^13 + 117766*x^12 - 327568*x^11 + 536090*x^10 - 646700*x^9 + 621936*x^8 - 488735*x^7 + 317878*x^6 - 166984*x^5 + 68952*x^4 - 21502*x^3 + 5711*x^2 - 1133*x + 193)
 
gp: K = bnfinit(x^28 - 12*x^27 + 74*x^26 - 288*x^25 + 773*x^24 - 1407*x^23 + 1601*x^22 - 676*x^21 - 794*x^20 + 2173*x^19 - 6644*x^18 + 23336*x^17 - 53512*x^16 + 82551*x^15 - 80880*x^14 + 23133*x^13 + 117766*x^12 - 327568*x^11 + 536090*x^10 - 646700*x^9 + 621936*x^8 - 488735*x^7 + 317878*x^6 - 166984*x^5 + 68952*x^4 - 21502*x^3 + 5711*x^2 - 1133*x + 193, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![193, -1133, 5711, -21502, 68952, -166984, 317878, -488735, 621936, -646700, 536090, -327568, 117766, 23133, -80880, 82551, -53512, 23336, -6644, 2173, -794, -676, 1601, -1407, 773, -288, 74, -12, 1]);
 

\( x^{28} - 12 x^{27} + 74 x^{26} - 288 x^{25} + 773 x^{24} - 1407 x^{23} + 1601 x^{22} - 676 x^{21} - 794 x^{20} + 2173 x^{19} - 6644 x^{18} + 23336 x^{17} - 53512 x^{16} + 82551 x^{15} - 80880 x^{14} + 23133 x^{13} + 117766 x^{12} - 327568 x^{11} + 536090 x^{10} - 646700 x^{9} + 621936 x^{8} - 488735 x^{7} + 317878 x^{6} - 166984 x^{5} + 68952 x^{4} - 21502 x^{3} + 5711 x^{2} - 1133 x + 193 \)

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1923732503543401200540313355166872100640336877\)\(\medspace = 43^{14}\cdot 53^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $41.43$
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:  $43, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{7} a^{20} + \frac{3}{7} a^{19} + \frac{3}{7} a^{18} + \frac{3}{7} a^{17} + \frac{3}{7} a^{16} - \frac{2}{7} a^{15} - \frac{3}{7} a^{13} - \frac{2}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{21} + \frac{1}{7} a^{19} + \frac{1}{7} a^{18} + \frac{1}{7} a^{17} + \frac{3}{7} a^{16} - \frac{1}{7} a^{15} - \frac{3}{7} a^{14} + \frac{3}{7} a^{11} - \frac{1}{7} a^{10} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{3}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{22} - \frac{2}{7} a^{19} - \frac{2}{7} a^{18} + \frac{3}{7} a^{16} - \frac{1}{7} a^{15} + \frac{3}{7} a^{13} - \frac{2}{7} a^{12} - \frac{2}{7} a^{11} + \frac{1}{7} a^{10} - \frac{1}{7} a^{9} - \frac{1}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{371} a^{23} - \frac{19}{371} a^{22} - \frac{3}{53} a^{21} + \frac{8}{371} a^{20} - \frac{172}{371} a^{19} - \frac{72}{371} a^{18} + \frac{40}{371} a^{17} + \frac{3}{53} a^{16} + \frac{111}{371} a^{15} + \frac{136}{371} a^{14} + \frac{44}{371} a^{13} - \frac{96}{371} a^{12} - \frac{19}{53} a^{11} + \frac{47}{371} a^{10} + \frac{146}{371} a^{9} + \frac{163}{371} a^{8} + \frac{103}{371} a^{7} + \frac{131}{371} a^{6} + \frac{13}{371} a^{5} + \frac{135}{371} a^{4} + \frac{163}{371} a^{3} - \frac{127}{371} a^{2} - \frac{65}{371} a - \frac{93}{371}$, $\frac{1}{2597} a^{24} - \frac{2}{2597} a^{23} + \frac{80}{2597} a^{22} + \frac{128}{2597} a^{21} + \frac{176}{2597} a^{20} - \frac{876}{2597} a^{19} - \frac{177}{2597} a^{18} - \frac{412}{2597} a^{17} + \frac{468}{2597} a^{16} - \frac{415}{2597} a^{15} + \frac{1296}{2597} a^{14} - \frac{81}{371} a^{13} - \frac{440}{2597} a^{12} + \frac{436}{2597} a^{11} - \frac{433}{2597} a^{10} - \frac{482}{2597} a^{9} + \frac{489}{2597} a^{8} - \frac{874}{2597} a^{7} + \frac{703}{2597} a^{6} - \frac{969}{2597} a^{5} - \frac{563}{2597} a^{4} + \frac{153}{2597} a^{3} - \frac{263}{2597} a^{2} + \frac{339}{2597} a - \frac{1263}{2597}$, $\frac{1}{2597} a^{25} - \frac{1}{2597} a^{23} - \frac{104}{2597} a^{22} - \frac{177}{2597} a^{21} - \frac{27}{2597} a^{20} + \frac{556}{2597} a^{19} - \frac{787}{2597} a^{18} + \frac{274}{2597} a^{17} + \frac{388}{2597} a^{16} - \frac{1032}{2597} a^{15} + \frac{828}{2597} a^{14} - \frac{881}{2597} a^{13} + \frac{641}{2597} a^{12} + \frac{1034}{2597} a^{11} - \frac{515}{2597} a^{10} + \frac{1268}{2597} a^{9} + \frac{167}{2597} a^{8} - \frac{1185}{2597} a^{7} - \frac{1117}{2597} a^{6} + \frac{950}{2597} a^{5} + \frac{19}{371} a^{4} + \frac{9}{49} a^{3} + \frac{1059}{2597} a^{2} - \frac{774}{2597} a + \frac{925}{2597}$, $\frac{1}{17979031} a^{26} - \frac{166}{2568433} a^{25} - \frac{709}{17979031} a^{24} + \frac{957}{781697} a^{23} - \frac{162853}{17979031} a^{22} - \frac{1160055}{17979031} a^{21} - \frac{1221526}{17979031} a^{20} + \frac{2436201}{17979031} a^{19} - \frac{2734302}{17979031} a^{18} - \frac{2754855}{17979031} a^{17} + \frac{4849983}{17979031} a^{16} - \frac{1811666}{17979031} a^{15} - \frac{8}{2568433} a^{14} + \frac{4179382}{17979031} a^{13} - \frac{8048323}{17979031} a^{12} + \frac{5315381}{17979031} a^{11} - \frac{395497}{2568433} a^{10} + \frac{312464}{17979031} a^{9} - \frac{2424409}{17979031} a^{8} - \frac{8946880}{17979031} a^{7} + \frac{6887253}{17979031} a^{6} + \frac{863936}{17979031} a^{5} + \frac{326678}{781697} a^{4} + \frac{3390761}{17979031} a^{3} + \frac{12603}{59731} a^{2} - \frac{8732278}{17979031} a - \frac{2330374}{17979031}$, $\frac{1}{615518663136640948199250770635903489437240592847} a^{27} + \frac{8331027850140854056708474856588623159476}{615518663136640948199250770635903489437240592847} a^{26} - \frac{44188593237243918795441882229418513252409466}{615518663136640948199250770635903489437240592847} a^{25} - \frac{6595093562231907922835763816195762977389345}{87931237590948706885607252947986212776748656121} a^{24} + \frac{11651822428737709333460130870522517954146726}{11613559681823414116966995672375537536551709299} a^{23} - \frac{15171220627885975314959933088589404146817053834}{615518663136640948199250770635903489437240592847} a^{22} - \frac{15185620733117072305933268703888728500770030414}{615518663136640948199250770635903489437240592847} a^{21} + \frac{28773566776874405887153350323122233923021168556}{615518663136640948199250770635903489437240592847} a^{20} - \frac{6069315318782191245522114360990188444620173702}{12561605370135529555086750421140887539535522303} a^{19} + \frac{31922588785346845076239546482656489479258580300}{615518663136640948199250770635903489437240592847} a^{18} + \frac{1994615116372558285491699603161224505063379708}{19855440746343256393524218407609789981846470737} a^{17} - \frac{289734478915455870307056073345521208305426624841}{615518663136640948199250770635903489437240592847} a^{16} + \frac{95790011893026361179691019276008588807705819214}{615518663136640948199250770635903489437240592847} a^{15} - \frac{16861327438320004945518205451553667120571611213}{615518663136640948199250770635903489437240592847} a^{14} + \frac{171955674354031904870769664115988205230078927961}{615518663136640948199250770635903489437240592847} a^{13} + \frac{29080744131601035545881789970437607437319956377}{87931237590948706885607252947986212776748656121} a^{12} - \frac{73653410184302212373059955353590816987483639220}{615518663136640948199250770635903489437240592847} a^{11} - \frac{121046836633973624073025302375675534897695153379}{615518663136640948199250770635903489437240592847} a^{10} - \frac{154236664733222941224952642281084436392038941095}{615518663136640948199250770635903489437240592847} a^{9} + \frac{128105986289056202512498213548104415379511648658}{615518663136640948199250770635903489437240592847} a^{8} + \frac{245194515113471411329482992024445492977691055962}{615518663136640948199250770635903489437240592847} a^{7} + \frac{1629240195043485151563805071520926955325988659}{26761681005940910791271772636343629975532199689} a^{6} - \frac{116528743250765722088132889158088792419874046841}{615518663136640948199250770635903489437240592847} a^{5} - \frac{271960484478823424584275716605543239605648728395}{615518663136640948199250770635903489437240592847} a^{4} + \frac{193054928828723969493181151690572869421362962085}{615518663136640948199250770635903489437240592847} a^{3} - \frac{98141900283845647559262841369666139644490733829}{615518663136640948199250770635903489437240592847} a^{2} + \frac{739118940701361655857273465912593485797576065}{1794515052876504222155250060162983934219360329} a - \frac{269578715783894575356818332355757453292888303209}{615518663136640948199250770635903489437240592847}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $13$
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:  \( 71584737217.10019 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 71584737217.10019 \cdot 3}{2\sqrt{1923732503543401200540313355166872100640336877}}\approx 0.365895871823473$ (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

\(\Q(\sqrt{-43}) \), 4.0.97997.1, 7.1.11836763639.1, 14.0.6024685858158758459803.1

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 $28$ $28$ $28$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/59.7.0.1}{7} }^{4}$

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
$43$43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
$53$$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{53}$$x + 2$$1$$1$$0$Trivial$[\ ]$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$
53.2.1.1$x^{2} - 53$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.43.2t1.a.a$1$ $ 43 $ \(\Q(\sqrt{-43}) \) $C_2$ (as 2T1) $1$ $-1$
1.2279.2t1.a.a$1$ $ 43 \cdot 53 $ \(\Q(\sqrt{-2279}) \) $C_2$ (as 2T1) $1$ $-1$
1.53.2t1.a.a$1$ $ 53 $ \(\Q(\sqrt{53}) \) $C_2$ (as 2T1) $1$ $1$
* 2.2279.4t3.a.a$2$ $ 43 \cdot 53 $ 4.0.97997.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.2279.14t3.a.a$2$ $ 43 \cdot 53 $ 14.0.6024685858158758459803.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.2279.7t2.a.b$2$ $ 43 \cdot 53 $ 7.1.11836763639.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.2279.14t3.a.c$2$ $ 43 \cdot 53 $ 14.0.6024685858158758459803.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.2279.7t2.a.c$2$ $ 43 \cdot 53 $ 7.1.11836763639.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.2279.7t2.a.a$2$ $ 43 \cdot 53 $ 7.1.11836763639.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.2279.14t3.a.b$2$ $ 43 \cdot 53 $ 14.0.6024685858158758459803.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.2279.28t10.a.c$2$ $ 43 \cdot 53 $ 28.0.1923732503543401200540313355166872100640336877.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2279.28t10.a.f$2$ $ 43 \cdot 53 $ 28.0.1923732503543401200540313355166872100640336877.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2279.28t10.a.d$2$ $ 43 \cdot 53 $ 28.0.1923732503543401200540313355166872100640336877.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2279.28t10.a.e$2$ $ 43 \cdot 53 $ 28.0.1923732503543401200540313355166872100640336877.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2279.28t10.a.b$2$ $ 43 \cdot 53 $ 28.0.1923732503543401200540313355166872100640336877.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.2279.28t10.a.a$2$ $ 43 \cdot 53 $ 28.0.1923732503543401200540313355166872100640336877.1 $D_{28}$ (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 *.