Properties

Label 28.0.537...517.1
Degree $28$
Signature $[0, 14]$
Discriminant $5.378\times 10^{47}$
Root discriminant $50.66$
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 - 24*x^26 + 286*x^24 - 2327*x^22 + 12825*x^20 - 40336*x^18 + 39135*x^16 + 152588*x^14 - 531606*x^12 + 381077*x^10 + 2209555*x^8 - 10321444*x^6 + 18143914*x^4 - 8922944*x^2 + 1423325)
 
gp: K = bnfinit(x^28 - 24*x^26 + 286*x^24 - 2327*x^22 + 12825*x^20 - 40336*x^18 + 39135*x^16 + 152588*x^14 - 531606*x^12 + 381077*x^10 + 2209555*x^8 - 10321444*x^6 + 18143914*x^4 - 8922944*x^2 + 1423325, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1423325, 0, -8922944, 0, 18143914, 0, -10321444, 0, 2209555, 0, 381077, 0, -531606, 0, 152588, 0, 39135, 0, -40336, 0, 12825, 0, -2327, 0, 286, 0, -24, 0, 1]);
 

\( x^{28} - 24 x^{26} + 286 x^{24} - 2327 x^{22} + 12825 x^{20} - 40336 x^{18} + 39135 x^{16} + 152588 x^{14} - 531606 x^{12} + 381077 x^{10} + 2209555 x^{8} - 10321444 x^{6} + 18143914 x^{4} - 8922944 x^{2} + 1423325 \)

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:  \(537788304782666838980279974392697944146481853517\)\(\medspace = 19^{14}\cdot 197^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $50.66$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{19} - \frac{1}{10} a^{17} - \frac{1}{5} a^{15} + \frac{3}{10} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{2}{5} a^{7} + \frac{1}{10} a^{5} - \frac{1}{2} a^{4} - \frac{3}{10} a^{3} - \frac{1}{2} a^{2} - \frac{3}{10} a - \frac{1}{2}$, $\frac{1}{190} a^{20} + \frac{2}{95} a^{18} + \frac{1}{5} a^{16} - \frac{27}{190} a^{14} - \frac{5}{38} a^{12} - \frac{1}{2} a^{11} - \frac{3}{38} a^{10} - \frac{23}{95} a^{8} + \frac{33}{95} a^{6} - \frac{1}{2} a^{5} - \frac{13}{190} a^{4} - \frac{1}{2} a^{3} - \frac{63}{190} a^{2} - \frac{13}{38}$, $\frac{1}{190} a^{21} + \frac{2}{95} a^{19} + \frac{1}{5} a^{17} - \frac{27}{190} a^{15} - \frac{5}{38} a^{13} - \frac{1}{2} a^{12} - \frac{3}{38} a^{11} - \frac{23}{95} a^{9} + \frac{33}{95} a^{7} - \frac{1}{2} a^{6} - \frac{13}{190} a^{5} - \frac{1}{2} a^{4} - \frac{63}{190} a^{3} - \frac{13}{38} a$, $\frac{1}{190} a^{22} + \frac{11}{95} a^{18} + \frac{11}{190} a^{16} - \frac{6}{95} a^{14} - \frac{1}{2} a^{13} + \frac{17}{38} a^{12} - \frac{1}{2} a^{11} - \frac{81}{190} a^{10} - \frac{7}{38} a^{8} - \frac{1}{2} a^{7} - \frac{87}{190} a^{6} + \frac{42}{95} a^{4} - \frac{3}{190} a^{2} - \frac{5}{38}$, $\frac{1}{190} a^{23} + \frac{3}{190} a^{19} + \frac{3}{19} a^{17} + \frac{13}{95} a^{15} + \frac{14}{95} a^{13} - \frac{1}{2} a^{12} - \frac{81}{190} a^{11} - \frac{7}{38} a^{9} + \frac{27}{190} a^{7} - \frac{3}{19} a^{5} + \frac{27}{95} a^{3} - \frac{1}{2} a^{2} + \frac{16}{95} a$, $\frac{1}{334970} a^{24} - \frac{92}{167485} a^{22} - \frac{436}{167485} a^{20} + \frac{79787}{334970} a^{18} - \frac{37574}{167485} a^{16} - \frac{27872}{167485} a^{14} - \frac{1}{2} a^{13} + \frac{51242}{167485} a^{12} + \frac{76412}{167485} a^{10} + \frac{64541}{167485} a^{8} - \frac{143707}{334970} a^{6} - \frac{2536}{167485} a^{4} + \frac{39369}{334970} a^{2} - \frac{1}{2} a - \frac{6553}{66994}$, $\frac{1}{5694490} a^{25} + \frac{12157}{5694490} a^{23} - \frac{3962}{2847245} a^{21} + \frac{10804}{2847245} a^{19} - \frac{91081}{569449} a^{17} - \frac{917851}{5694490} a^{15} + \frac{42335}{569449} a^{13} + \frac{598863}{5694490} a^{11} - \frac{1}{2} a^{10} + \frac{261997}{2847245} a^{9} - \frac{1}{2} a^{8} + \frac{397986}{2847245} a^{7} - \frac{1}{2} a^{6} + \frac{98926}{569449} a^{5} - \frac{1}{2} a^{4} + \frac{1686011}{5694490} a^{3} - \frac{410413}{2847245} a - \frac{1}{2}$, $\frac{1}{253552965241621004964380111454148405250} a^{26} - \frac{340363725244844817005532862612233}{253552965241621004964380111454148405250} a^{24} - \frac{419613924332448198636226769125933117}{253552965241621004964380111454148405250} a^{22} + \frac{2633447384449397485594035049825188}{126776482620810502482190055727074202625} a^{20} + \frac{25056292054960139534428471592787307483}{126776482620810502482190055727074202625} a^{18} - \frac{4448161023512888831126016279933850238}{25355296524162100496438011145414840525} a^{16} + \frac{556354825217861600438421538064986578}{25355296524162100496438011145414840525} a^{14} - \frac{1}{2} a^{13} + \frac{55057370409198503764650882195858946484}{126776482620810502482190055727074202625} a^{12} + \frac{81017613544042945187289612317669711057}{253552965241621004964380111454148405250} a^{10} - \frac{1}{2} a^{9} - \frac{3109647191674111385461500038973138419}{13344892907453737103388426918639389750} a^{8} + \frac{100422110159481816919766471295153459429}{253552965241621004964380111454148405250} a^{6} - \frac{1}{2} a^{5} + \frac{2028070283834097855820065459792680809}{50710593048324200992876022290829681050} a^{4} - \frac{99679688775783833707458355095145573091}{253552965241621004964380111454148405250} a^{2} - \frac{1}{2} a - \frac{125224855574878995984347408367887469}{596595212333225894033835556362702130}$, $\frac{1}{253552965241621004964380111454148405250} a^{27} + \frac{15844420161147298129455677473567}{253552965241621004964380111454148405250} a^{25} - \frac{46329686433961592978349581447339721}{126776482620810502482190055727074202625} a^{23} - \frac{74173933968717652340387868906175437}{126776482620810502482190055727074202625} a^{21} + \frac{5764447376783381989478910577054961341}{253552965241621004964380111454148405250} a^{19} + \frac{8687527937591981959468663358214228579}{50710593048324200992876022290829681050} a^{17} - \frac{166151648801748138913528154618944471}{2668978581490747420677685383727877950} a^{15} + \frac{56738597691981591789856230981858659793}{253552965241621004964380111454148405250} a^{13} + \frac{871680250876026726912042986070662776}{3092109332214890304443659895782297625} a^{11} - \frac{1}{2} a^{10} + \frac{18139512460938671806489583600386059289}{253552965241621004964380111454148405250} a^{9} - \frac{1}{2} a^{8} + \frac{122393919088486925561580401918995817929}{253552965241621004964380111454148405250} a^{7} + \frac{1689009504045045811977883327148171752}{25355296524162100496438011145414840525} a^{5} - \frac{55819710891904097361064993601046909408}{126776482620810502482190055727074202625} a^{3} - \frac{2294776469627856106050399325367152049}{5071059304832420099287602229082968105} a$

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:  $13$
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:  \( 337496841957.8353 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 337496841957.8353 \cdot 29}{2\sqrt{537788304782666838980279974392697944146481853517}}\approx 0.997355979843452$ (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

\(\Q(\sqrt{-19}) \), 4.0.71117.1, 7.1.52439613407.1, 14.0.52248348031236668805331.1

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

Sibling fields

Degree 28 sibling: 28.2.5576015581167650909427113418703236578781943428571.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $28$ $28$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $28$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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$19.2.1.2$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$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
$197$$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{197}$$x + 2$$1$$1$$0$Trivial$[\ ]$
197.2.1.1$x^{2} - 197$$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.c.a$2$ $ 19 \cdot 197 $ 4.2.737371.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.3743.7t2.a.a$2$ $ 19 \cdot 197 $ 7.1.52439613407.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3743.14t3.b.c$2$ $ 19 \cdot 197 $ 14.2.541732871692295987086853.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3743.14t3.b.a$2$ $ 19 \cdot 197 $ 14.2.541732871692295987086853.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3743.7t2.a.c$2$ $ 19 \cdot 197 $ 7.1.52439613407.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3743.7t2.a.b$2$ $ 19 \cdot 197 $ 7.1.52439613407.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3743.14t3.b.b$2$ $ 19 \cdot 197 $ 14.2.541732871692295987086853.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3743.28t10.b.c$2$ $ 19 \cdot 197 $ 28.0.537788304782666838980279974392697944146481853517.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3743.28t10.b.a$2$ $ 19 \cdot 197 $ 28.0.537788304782666838980279974392697944146481853517.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3743.28t10.b.d$2$ $ 19 \cdot 197 $ 28.0.537788304782666838980279974392697944146481853517.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3743.28t10.b.f$2$ $ 19 \cdot 197 $ 28.0.537788304782666838980279974392697944146481853517.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3743.28t10.b.e$2$ $ 19 \cdot 197 $ 28.0.537788304782666838980279974392697944146481853517.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3743.28t10.b.b$2$ $ 19 \cdot 197 $ 28.0.537788304782666838980279974392697944146481853517.1 $D_{28}$ (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 *.