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Properties

Label 28.2.557...571.1
Degree $28$
Signature $[2, 13]$
Discriminant $-5.576\times 10^{48}$
Root discriminant $55.07$
Ramified primes $19, 197$
Class number $29$ (GRH)
Class group $[29]$ (GRH)
Galois group $D_{28}$ (as 28T10)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 20*x^26 + 294*x^24 - 3361*x^22 + 28147*x^20 - 154673*x^18 + 536790*x^16 - 1083403*x^14 + 980940*x^12 + 64462*x^10 - 677733*x^8 - 44521*x^6 + 115444*x^4 + 5415*x^2 - 6859)
 
gp: K = bnfinit(x^28 - 20*x^26 + 294*x^24 - 3361*x^22 + 28147*x^20 - 154673*x^18 + 536790*x^16 - 1083403*x^14 + 980940*x^12 + 64462*x^10 - 677733*x^8 - 44521*x^6 + 115444*x^4 + 5415*x^2 - 6859, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6859, 0, 5415, 0, 115444, 0, -44521, 0, -677733, 0, 64462, 0, 980940, 0, -1083403, 0, 536790, 0, -154673, 0, 28147, 0, -3361, 0, 294, 0, -20, 0, 1]);
 

\( x^{28} - 20 x^{26} + 294 x^{24} - 3361 x^{22} + 28147 x^{20} - 154673 x^{18} + 536790 x^{16} - 1083403 x^{14} + 980940 x^{12} + 64462 x^{10} - 677733 x^{8} - 44521 x^{6} + 115444 x^{4} + 5415 x^{2} - 6859 \)

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:  \(-5576015581167650909427113418703236578781943428571\)\(\medspace = -\,19^{13}\cdot 197^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $55.07$
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:  $19, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{10} a^{14} + \frac{1}{5} a^{12} - \frac{1}{2} a^{11} + \frac{1}{10} a^{10} - \frac{1}{2} a^{9} + \frac{2}{5} a^{8} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{1}{2} a + \frac{1}{10}$, $\frac{1}{10} a^{15} + \frac{1}{5} a^{13} - \frac{1}{2} a^{12} + \frac{1}{10} a^{11} - \frac{1}{2} a^{10} + \frac{2}{5} a^{9} - \frac{1}{2} a^{8} + \frac{2}{5} a^{7} + \frac{1}{10} a^{5} - \frac{1}{2} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} + \frac{1}{10} a$, $\frac{1}{10} a^{16} - \frac{1}{2} a^{13} - \frac{3}{10} a^{12} - \frac{1}{2} a^{11} + \frac{1}{5} a^{10} - \frac{1}{2} a^{9} - \frac{2}{5} a^{8} + \frac{3}{10} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{3}{10} a^{2} - \frac{1}{5}$, $\frac{1}{10} a^{17} - \frac{3}{10} a^{13} - \frac{1}{2} a^{12} - \frac{3}{10} a^{11} + \frac{1}{10} a^{9} - \frac{1}{5} a^{7} - \frac{1}{2} a^{6} + \frac{1}{5} a^{3} + \frac{3}{10} a - \frac{1}{2}$, $\frac{1}{10} a^{18} - \frac{1}{2} a^{13} + \frac{3}{10} a^{12} - \frac{1}{2} a^{11} + \frac{2}{5} a^{10} - \frac{1}{2} a^{9} + \frac{1}{5} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{10} a^{2} + \frac{3}{10}$, $\frac{1}{10} a^{19} + \frac{3}{10} a^{13} - \frac{1}{2} a^{12} - \frac{1}{10} a^{11} - \frac{1}{2} a^{9} - \frac{3}{10} a^{7} - \frac{1}{2} a^{5} + \frac{2}{5} a^{3} - \frac{1}{5} a - \frac{1}{2}$, $\frac{1}{10} a^{20} - \frac{1}{2} a^{13} + \frac{3}{10} a^{12} - \frac{1}{2} a^{11} + \frac{1}{5} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{3}{10} a^{6} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} - \frac{3}{10}$, $\frac{1}{10} a^{21} + \frac{3}{10} a^{13} - \frac{1}{2} a^{12} - \frac{3}{10} a^{11} - \frac{1}{2} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{5} - \frac{3}{10} a^{3} + \frac{1}{5} a - \frac{1}{2}$, $\frac{1}{89870} a^{22} - \frac{687}{89870} a^{20} + \frac{509}{89870} a^{18} + \frac{447}{89870} a^{16} + \frac{1457}{44935} a^{14} + \frac{12137}{44935} a^{12} + \frac{2197}{8170} a^{10} - \frac{1}{2} a^{9} - \frac{8571}{89870} a^{8} - \frac{3807}{89870} a^{6} - \frac{1}{2} a^{5} - \frac{4812}{44935} a^{4} - \frac{2531}{8987} a^{2} + \frac{293}{946}$, $\frac{1}{89870} a^{23} - \frac{687}{89870} a^{21} + \frac{509}{89870} a^{19} + \frac{447}{89870} a^{17} + \frac{1457}{44935} a^{15} + \frac{12137}{44935} a^{13} + \frac{2197}{8170} a^{11} - \frac{1}{2} a^{10} - \frac{8571}{89870} a^{9} - \frac{3807}{89870} a^{7} - \frac{1}{2} a^{6} - \frac{4812}{44935} a^{5} - \frac{2531}{8987} a^{3} + \frac{293}{946} a$, $\frac{1}{7638950} a^{24} + \frac{3}{7638950} a^{22} + \frac{118226}{3819475} a^{20} + \frac{351657}{7638950} a^{18} + \frac{263}{347225} a^{16} - \frac{13537}{694450} a^{14} - \frac{1}{2} a^{13} - \frac{13378}{224675} a^{12} - \frac{1}{2} a^{11} + \frac{1759339}{3819475} a^{10} + \frac{122}{201025} a^{8} - \frac{1}{2} a^{7} - \frac{3508193}{7638950} a^{6} + \frac{1636876}{3819475} a^{4} - \frac{3200657}{7638950} a^{2} - \frac{1}{2} a + \frac{29587}{201025}$, $\frac{1}{7638950} a^{25} + \frac{3}{7638950} a^{23} + \frac{118226}{3819475} a^{21} + \frac{351657}{7638950} a^{19} + \frac{263}{347225} a^{17} - \frac{13537}{694450} a^{15} - \frac{13378}{224675} a^{13} - \frac{1}{2} a^{12} - \frac{300797}{7638950} a^{11} - \frac{1}{2} a^{10} - \frac{200781}{402050} a^{9} - \frac{1}{2} a^{8} + \frac{155641}{3819475} a^{7} + \frac{1636876}{3819475} a^{5} - \frac{1}{2} a^{4} + \frac{309409}{3819475} a^{3} - \frac{1}{2} a^{2} - \frac{141851}{402050} a - \frac{1}{2}$, $\frac{1}{801968702694067864307577420250} a^{26} - \frac{7961464319335167599091}{801968702694067864307577420250} a^{24} + \frac{165110168513700952411158}{80196870269406786430757742025} a^{22} + \frac{3069872686612919128317897567}{400984351347033932153788710125} a^{20} - \frac{16443039159492839840162159321}{400984351347033932153788710125} a^{18} - \frac{14475131372525622588698825833}{400984351347033932153788710125} a^{16} - \frac{2321865883091506467997257849}{801968702694067864307577420250} a^{14} - \frac{1}{2} a^{13} - \frac{1828975306673685621459496859}{72906245699460714937052492750} a^{12} - \frac{3023951627464658996135931396}{9325217473186835631483458375} a^{10} - \frac{6360990085335086781064735411}{47174629570239286135739848250} a^{8} - \frac{3551664313508888132629141353}{400984351347033932153788710125} a^{6} - \frac{4160749943052043243011354312}{80196870269406786430757742025} a^{4} + \frac{198475908342842901576951528}{21104439544580733271252037375} a^{2} + \frac{129533876773782101989931243}{1110759976030564909013265125}$, $\frac{1}{15237405351187289421843970984750} a^{27} + \frac{468447960543296225282082}{7618702675593644710921985492375} a^{25} + \frac{6695124543208000713494987}{1523740535118728942184397098475} a^{23} - \frac{134148164893227132834068638381}{15237405351187289421843970984750} a^{21} + \frac{250977740930978956620335102093}{15237405351187289421843970984750} a^{19} - \frac{316348688489665522705648235661}{15237405351187289421843970984750} a^{17} + \frac{99700335791227439177841007883}{7618702675593644710921985492375} a^{15} + \frac{573298577611301559306757254583}{7618702675593644710921985492375} a^{13} + \frac{161880199778947245071135076592}{7618702675593644710921985492375} a^{11} - \frac{1}{2} a^{10} + \frac{3290497756484872124590170499643}{15237405351187289421843970984750} a^{9} - \frac{1}{2} a^{8} - \frac{564195322185700913600426059311}{1385218668289753583803997362250} a^{7} - \frac{1}{2} a^{6} - \frac{424153802117684074796242132077}{3047481070237457884368794196950} a^{5} + \frac{28156513694559321251457327441}{801968702694067864307577420250} a^{3} - \frac{1}{2} a^{2} + \frac{4463743898903588100150205691}{42208879089161466542504074750} a - \frac{1}{2}$

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

Class group and class number

$C_{29}$, which has order $29$ (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:  \( 1125038158310.573 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{13}\cdot 1125038158310.573 \cdot 29}{2\sqrt{5576015581167650909427113418703236578781943428571}}\approx 0.657311976996382$ (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

\(\Q(\sqrt{197}) \), 4.2.737371.1, 7.1.52439613407.1, 14.2.541732871692295987086853.1

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

Sibling fields

Degree 28 sibling: 28.0.537788304782666838980279974392697944146481853517.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $28$ $28$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ R ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $28$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
$197$197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$
197.2.1.1$x^{2} - 197$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.197.2t1.a.a$1$ $ 197 $ \(\Q(\sqrt{197}) \) $C_2$ (as 2T1) $1$ $1$
1.3743.2t1.a.a$1$ $ 19 \cdot 197 $ \(\Q(\sqrt{-3743}) \) $C_2$ (as 2T1) $1$ $-1$
1.19.2t1.a.a$1$ $ 19 $ \(\Q(\sqrt{-19}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3743.4t3.a.a$2$ $ 19 \cdot 197 $ 4.0.71117.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.3743.14t3.a.b$2$ $ 19 \cdot 197 $ 14.0.52248348031236668805331.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3743.7t2.a.c$2$ $ 19 \cdot 197 $ 7.1.52439613407.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3743.14t3.a.a$2$ $ 19 \cdot 197 $ 14.0.52248348031236668805331.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3743.7t2.a.a$2$ $ 19 \cdot 197 $ 7.1.52439613407.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3743.7t2.a.b$2$ $ 19 \cdot 197 $ 7.1.52439613407.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3743.14t3.a.c$2$ $ 19 \cdot 197 $ 14.0.52248348031236668805331.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3743.28t10.a.b$2$ $ 19 \cdot 197 $ 28.2.5576015581167650909427113418703236578781943428571.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3743.28t10.a.d$2$ $ 19 \cdot 197 $ 28.2.5576015581167650909427113418703236578781943428571.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3743.28t10.a.e$2$ $ 19 \cdot 197 $ 28.2.5576015581167650909427113418703236578781943428571.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3743.28t10.a.a$2$ $ 19 \cdot 197 $ 28.2.5576015581167650909427113418703236578781943428571.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3743.28t10.a.f$2$ $ 19 \cdot 197 $ 28.2.5576015581167650909427113418703236578781943428571.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3743.28t10.a.c$2$ $ 19 \cdot 197 $ 28.2.5576015581167650909427113418703236578781943428571.1 $D_{28}$ (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 *.