Properties

Label 28.2.271...000.1
Degree $28$
Signature $[2, 13]$
Discriminant $-2.716\times 10^{51}$
Root discriminant $68.70$
Ramified primes $2, 5, 151$
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 - 15*x^26 - 390*x^24 + 2850*x^22 + 49825*x^20 + 260250*x^18 + 737875*x^16 + 549375*x^14 - 1364375*x^12 + 4565625*x^10 + 13346875*x^8 - 16781250*x^6 + 7031250*x^4 + 128437500*x^2 - 11796875)
 
gp: K = bnfinit(x^28 - 15*x^26 - 390*x^24 + 2850*x^22 + 49825*x^20 + 260250*x^18 + 737875*x^16 + 549375*x^14 - 1364375*x^12 + 4565625*x^10 + 13346875*x^8 - 16781250*x^6 + 7031250*x^4 + 128437500*x^2 - 11796875, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11796875, 0, 128437500, 0, 7031250, 0, -16781250, 0, 13346875, 0, 4565625, 0, -1364375, 0, 549375, 0, 737875, 0, 260250, 0, 49825, 0, 2850, 0, -390, 0, -15, 0, 1]);
 

\( x^{28} - 15 x^{26} - 390 x^{24} + 2850 x^{22} + 49825 x^{20} + 260250 x^{18} + 737875 x^{16} + 549375 x^{14} - 1364375 x^{12} + 4565625 x^{10} + 13346875 x^{8} - 16781250 x^{6} + 7031250 x^{4} + 128437500 x^{2} - 11796875 \)

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:  \(-2715878171109088684095776889728000000000000000000000\)\(\medspace = -\,2^{28}\cdot 5^{21}\cdot 151^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $68.70$
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:  $2, 5, 151$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{75} a^{8} - \frac{1}{3}$, $\frac{1}{75} a^{9} - \frac{1}{3} a$, $\frac{1}{75} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{75} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{375} a^{12} - \frac{1}{15} a^{4}$, $\frac{1}{375} a^{13} - \frac{1}{15} a^{5}$, $\frac{1}{375} a^{14} - \frac{1}{15} a^{6}$, $\frac{1}{375} a^{15} - \frac{1}{15} a^{7}$, $\frac{1}{39375} a^{16} + \frac{1}{875} a^{14} + \frac{1}{2625} a^{12} - \frac{1}{315} a^{8} + \frac{8}{105} a^{4} - \frac{1}{7} a^{2} - \frac{23}{63}$, $\frac{1}{39375} a^{17} + \frac{1}{875} a^{15} + \frac{1}{2625} a^{13} - \frac{1}{315} a^{9} + \frac{8}{105} a^{5} - \frac{1}{7} a^{3} - \frac{23}{63} a$, $\frac{1}{39375} a^{18} - \frac{1}{2625} a^{14} - \frac{1}{875} a^{12} - \frac{1}{315} a^{10} - \frac{2}{525} a^{8} + \frac{1}{105} a^{6} + \frac{1}{35} a^{4} + \frac{4}{63} a^{2} + \frac{2}{21}$, $\frac{1}{39375} a^{19} - \frac{1}{2625} a^{15} - \frac{1}{875} a^{13} - \frac{1}{315} a^{11} - \frac{2}{525} a^{9} + \frac{1}{105} a^{7} + \frac{1}{35} a^{5} + \frac{4}{63} a^{3} + \frac{2}{21} a$, $\frac{1}{590625} a^{20} - \frac{1}{118125} a^{16} - \frac{1}{2625} a^{14} + \frac{22}{23625} a^{12} - \frac{2}{525} a^{10} + \frac{2}{675} a^{8} + \frac{2}{21} a^{6} - \frac{32}{945} a^{4} - \frac{1}{3} a^{2} - \frac{4}{189}$, $\frac{1}{590625} a^{21} - \frac{1}{118125} a^{17} - \frac{1}{2625} a^{15} + \frac{22}{23625} a^{13} - \frac{2}{525} a^{11} + \frac{2}{675} a^{9} + \frac{2}{21} a^{7} - \frac{32}{945} a^{5} - \frac{1}{3} a^{3} - \frac{4}{189} a$, $\frac{1}{590625} a^{22} - \frac{1}{118125} a^{18} - \frac{2}{3375} a^{14} - \frac{2}{2625} a^{12} + \frac{2}{675} a^{10} - \frac{1}{175} a^{8} + \frac{31}{945} a^{6} + \frac{8}{105} a^{4} - \frac{31}{189} a^{2} - \frac{1}{7}$, $\frac{1}{590625} a^{23} - \frac{1}{118125} a^{19} - \frac{2}{3375} a^{15} - \frac{2}{2625} a^{13} + \frac{2}{675} a^{11} - \frac{1}{175} a^{9} + \frac{31}{945} a^{7} + \frac{8}{105} a^{5} - \frac{31}{189} a^{3} - \frac{1}{7} a$, $\frac{1}{611296875} a^{24} + \frac{13}{40753125} a^{22} - \frac{11}{17465625} a^{20} - \frac{64}{8150625} a^{18} - \frac{208}{24451875} a^{16} + \frac{2101}{1630125} a^{14} - \frac{1982}{4890375} a^{12} - \frac{83}{65205} a^{10} + \frac{683}{139725} a^{8} - \frac{610}{13041} a^{6} + \frac{4723}{195615} a^{4} - \frac{853}{13041} a^{2} - \frac{1763}{5589}$, $\frac{1}{611296875} a^{25} + \frac{13}{40753125} a^{23} - \frac{11}{17465625} a^{21} - \frac{64}{8150625} a^{19} - \frac{208}{24451875} a^{17} + \frac{2101}{1630125} a^{15} - \frac{1982}{4890375} a^{13} - \frac{83}{65205} a^{11} + \frac{683}{139725} a^{9} - \frac{610}{13041} a^{7} + \frac{4723}{195615} a^{5} - \frac{853}{13041} a^{3} - \frac{1763}{5589} a$, $\frac{1}{9799276971668751656578125} a^{26} + \frac{7428516806434652}{9799276971668751656578125} a^{24} + \frac{35230421701285726}{78394215773350013252625} a^{22} + \frac{1126679523245289119}{1959855394333750331315625} a^{20} - \frac{793210840901049751}{391971078866750066263125} a^{18} + \frac{38556257316441274}{30151621451288466635625} a^{16} + \frac{96148210534191883822}{78394215773350013252625} a^{14} - \frac{3737213328031508251}{78394215773350013252625} a^{12} + \frac{53208230574525667997}{15678843154670002650525} a^{10} - \frac{3700322039131097573}{1206064858051538665425} a^{8} + \frac{9327356486516156885}{627153726186800106021} a^{6} - \frac{44279557020158794159}{447966947276285790015} a^{4} - \frac{226901790574846654301}{627153726186800106021} a^{2} - \frac{59476367874564086701}{627153726186800106021}$, $\frac{1}{9799276971668751656578125} a^{27} + \frac{7428516806434652}{9799276971668751656578125} a^{25} + \frac{35230421701285726}{78394215773350013252625} a^{23} + \frac{1126679523245289119}{1959855394333750331315625} a^{21} - \frac{793210840901049751}{391971078866750066263125} a^{19} + \frac{38556257316441274}{30151621451288466635625} a^{17} + \frac{96148210534191883822}{78394215773350013252625} a^{15} - \frac{3737213328031508251}{78394215773350013252625} a^{13} + \frac{53208230574525667997}{15678843154670002650525} a^{11} - \frac{3700322039131097573}{1206064858051538665425} a^{9} + \frac{9327356486516156885}{627153726186800106021} a^{7} - \frac{44279557020158794159}{447966947276285790015} a^{5} - \frac{226901790574846654301}{627153726186800106021} a^{3} - \frac{59476367874564086701}{627153726186800106021} a$

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

\(\Q(\sqrt{5}) \), 4.2.302000.2, 7.1.3442951.1, 14.2.926086842843828125.1

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 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{14}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}$ $28$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $28$ $28$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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
2Data not computed
5Data not computed
$151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.755.2t1.a.a$1$ $ 5 \cdot 151 $ \(\Q(\sqrt{-755}) \) $C_2$ (as 2T1) $1$ $-1$
1.151.2t1.a.a$1$ $ 151 $ \(\Q(\sqrt{-151}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.60400.4t3.c.a$2$ $ 2^{4} \cdot 5^{2} \cdot 151 $ 4.0.45602000.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.151.7t2.a.a$2$ $ 151 $ 7.1.3442951.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3775.14t3.a.c$2$ $ 5^{2} \cdot 151 $ 14.0.139839113269418046875.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3775.14t3.a.a$2$ $ 5^{2} \cdot 151 $ 14.0.139839113269418046875.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.151.7t2.a.c$2$ $ 151 $ 7.1.3442951.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.151.7t2.a.b$2$ $ 151 $ 7.1.3442951.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3775.14t3.a.b$2$ $ 5^{2} \cdot 151 $ 14.0.139839113269418046875.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.60400.28t10.a.a$2$ $ 2^{4} \cdot 5^{2} \cdot 151 $ 28.2.2715878171109088684095776889728000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.60400.28t10.a.e$2$ $ 2^{4} \cdot 5^{2} \cdot 151 $ 28.2.2715878171109088684095776889728000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.60400.28t10.a.b$2$ $ 2^{4} \cdot 5^{2} \cdot 151 $ 28.2.2715878171109088684095776889728000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.60400.28t10.a.f$2$ $ 2^{4} \cdot 5^{2} \cdot 151 $ 28.2.2715878171109088684095776889728000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.60400.28t10.a.c$2$ $ 2^{4} \cdot 5^{2} \cdot 151 $ 28.2.2715878171109088684095776889728000000000000000000000.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.60400.28t10.a.d$2$ $ 2^{4} \cdot 5^{2} \cdot 151 $ 28.2.2715878171109088684095776889728000000000000000000000.1 $D_{28}$ (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 *.