Properties

Label 28.0.204...801.1
Degree $28$
Signature $[0, 14]$
Discriminant $2.044\times 10^{46}$
Root discriminant $45.08$
Ramified primes $7, 449$
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 - 6*x^27 + 18*x^26 + 6*x^25 - 69*x^24 + 47*x^23 + 257*x^22 - 674*x^21 + 321*x^20 + 545*x^19 - 15*x^18 - 692*x^17 + 1854*x^16 - 6800*x^15 + 5094*x^14 + 6261*x^13 - 5929*x^12 + 6262*x^11 - 14458*x^10 - 7839*x^9 + 27851*x^8 - 6775*x^7 + 6005*x^6 - 1487*x^5 - 35142*x^4 + 22011*x^3 + 18446*x^2 - 12525*x + 2675)
 
gp: K = bnfinit(x^28 - 6*x^27 + 18*x^26 + 6*x^25 - 69*x^24 + 47*x^23 + 257*x^22 - 674*x^21 + 321*x^20 + 545*x^19 - 15*x^18 - 692*x^17 + 1854*x^16 - 6800*x^15 + 5094*x^14 + 6261*x^13 - 5929*x^12 + 6262*x^11 - 14458*x^10 - 7839*x^9 + 27851*x^8 - 6775*x^7 + 6005*x^6 - 1487*x^5 - 35142*x^4 + 22011*x^3 + 18446*x^2 - 12525*x + 2675, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2675, -12525, 18446, 22011, -35142, -1487, 6005, -6775, 27851, -7839, -14458, 6262, -5929, 6261, 5094, -6800, 1854, -692, -15, 545, 321, -674, 257, 47, -69, 6, 18, -6, 1]);
 

\( x^{28} - 6 x^{27} + 18 x^{26} + 6 x^{25} - 69 x^{24} + 47 x^{23} + 257 x^{22} - 674 x^{21} + 321 x^{20} + 545 x^{19} - 15 x^{18} - 692 x^{17} + 1854 x^{16} - 6800 x^{15} + 5094 x^{14} + 6261 x^{13} - 5929 x^{12} + 6262 x^{11} - 14458 x^{10} - 7839 x^{9} + 27851 x^{8} - 6775 x^{7} + 6005 x^{6} - 1487 x^{5} - 35142 x^{4} + 22011 x^{3} + 18446 x^{2} - 12525 x + 2675 \)

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:  \(20444429365013571404907915184985572799247638801\)\(\medspace = 7^{14}\cdot 449^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $45.08$
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:  $7, 449$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{6} + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{2}{7} a^{7} + \frac{1}{7} a$, $\frac{1}{7} a^{14} - \frac{2}{7} a^{8} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{15} - \frac{2}{7} a^{9} + \frac{1}{7} a^{3}$, $\frac{1}{35} a^{16} + \frac{1}{35} a^{13} + \frac{1}{7} a^{10} + \frac{1}{7} a^{7} + \frac{1}{35} a^{4} + \frac{1}{35} a$, $\frac{1}{35} a^{17} + \frac{1}{35} a^{14} + \frac{1}{7} a^{11} + \frac{1}{7} a^{8} + \frac{1}{35} a^{5} + \frac{1}{35} a^{2}$, $\frac{1}{35} a^{18} + \frac{1}{35} a^{15} + \frac{1}{7} a^{9} + \frac{11}{35} a^{6} + \frac{1}{35} a^{3} - \frac{1}{7}$, $\frac{1}{245} a^{19} + \frac{3}{245} a^{18} - \frac{1}{245} a^{17} + \frac{3}{245} a^{15} - \frac{16}{245} a^{14} - \frac{6}{245} a^{13} - \frac{3}{49} a^{12} + \frac{20}{49} a^{11} - \frac{3}{7} a^{10} + \frac{3}{49} a^{9} - \frac{2}{49} a^{8} - \frac{19}{245} a^{7} + \frac{4}{35} a^{6} - \frac{106}{245} a^{5} + \frac{3}{7} a^{4} + \frac{3}{245} a^{3} - \frac{86}{245} a^{2} - \frac{46}{245} a + \frac{8}{49}$, $\frac{1}{245} a^{20} - \frac{3}{245} a^{18} + \frac{3}{245} a^{17} + \frac{3}{245} a^{16} + \frac{17}{245} a^{15} + \frac{1}{35} a^{14} + \frac{3}{245} a^{13} + \frac{1}{49} a^{12} + \frac{17}{49} a^{11} + \frac{17}{49} a^{10} - \frac{18}{49} a^{9} + \frac{81}{245} a^{8} + \frac{17}{49} a^{7} - \frac{78}{245} a^{6} - \frac{67}{245} a^{5} - \frac{67}{245} a^{4} - \frac{53}{245} a^{3} - \frac{68}{245} a^{2} - \frac{67}{245} a - \frac{10}{49}$, $\frac{1}{1225} a^{21} + \frac{2}{1225} a^{20} + \frac{2}{1225} a^{19} - \frac{16}{1225} a^{18} - \frac{17}{1225} a^{17} + \frac{2}{1225} a^{16} + \frac{4}{175} a^{15} - \frac{12}{175} a^{14} - \frac{3}{49} a^{13} + \frac{11}{245} a^{12} - \frac{66}{245} a^{11} + \frac{86}{245} a^{10} - \frac{409}{1225} a^{9} + \frac{582}{1225} a^{8} - \frac{38}{1225} a^{7} + \frac{274}{1225} a^{6} - \frac{17}{1225} a^{5} + \frac{317}{1225} a^{4} + \frac{58}{1225} a^{3} - \frac{409}{1225} a^{2} + \frac{4}{245} a + \frac{11}{49}$, $\frac{1}{1225} a^{22} - \frac{2}{1225} a^{20} + \frac{1}{245} a^{18} + \frac{16}{1225} a^{17} - \frac{11}{1225} a^{16} + \frac{1}{49} a^{15} - \frac{52}{1225} a^{14} + \frac{2}{49} a^{13} - \frac{8}{245} a^{12} - \frac{117}{245} a^{11} + \frac{131}{1225} a^{10} - \frac{9}{49} a^{9} - \frac{527}{1225} a^{8} - \frac{41}{245} a^{7} + \frac{11}{49} a^{6} - \frac{544}{1225} a^{5} + \frac{264}{1225} a^{4} - \frac{72}{245} a^{3} + \frac{74}{175} a^{2} + \frac{101}{245} a + \frac{3}{49}$, $\frac{1}{1225} a^{23} - \frac{1}{1225} a^{20} - \frac{1}{1225} a^{19} + \frac{4}{1225} a^{18} - \frac{3}{245} a^{17} + \frac{2}{175} a^{16} - \frac{76}{1225} a^{15} + \frac{6}{175} a^{14} + \frac{6}{245} a^{13} - \frac{554}{1225} a^{11} + \frac{1}{35} a^{10} + \frac{71}{245} a^{9} - \frac{396}{1225} a^{8} - \frac{386}{1225} a^{7} - \frac{201}{1225} a^{6} + \frac{87}{245} a^{5} - \frac{9}{25} a^{4} - \frac{321}{1225} a^{3} - \frac{303}{1225} a^{2} - \frac{4}{35} a + \frac{23}{49}$, $\frac{1}{8575} a^{24} + \frac{2}{8575} a^{23} - \frac{2}{8575} a^{22} - \frac{2}{8575} a^{21} - \frac{1}{8575} a^{20} + \frac{1}{1715} a^{19} + \frac{1}{175} a^{18} + \frac{69}{8575} a^{17} + \frac{1}{1225} a^{16} - \frac{138}{8575} a^{15} + \frac{327}{8575} a^{14} + \frac{78}{1715} a^{13} - \frac{254}{8575} a^{12} - \frac{2223}{8575} a^{11} - \frac{617}{8575} a^{10} - \frac{1027}{8575} a^{9} - \frac{208}{1225} a^{8} - \frac{229}{1715} a^{7} - \frac{138}{1225} a^{6} - \frac{2566}{8575} a^{5} + \frac{962}{8575} a^{4} + \frac{3442}{8575} a^{3} + \frac{3202}{8575} a^{2} + \frac{129}{1715} a - \frac{152}{343}$, $\frac{1}{317275} a^{25} + \frac{13}{317275} a^{24} + \frac{34}{317275} a^{23} + \frac{88}{317275} a^{22} - \frac{1}{8575} a^{21} + \frac{78}{317275} a^{20} + \frac{587}{317275} a^{19} - \frac{1212}{317275} a^{18} + \frac{4336}{317275} a^{17} + \frac{818}{63455} a^{16} + \frac{4493}{317275} a^{15} + \frac{3777}{317275} a^{14} - \frac{7654}{317275} a^{13} - \frac{18842}{317275} a^{12} - \frac{48226}{317275} a^{11} - \frac{157572}{317275} a^{10} + \frac{96118}{317275} a^{9} + \frac{153723}{317275} a^{8} - \frac{96343}{317275} a^{7} + \frac{105423}{317275} a^{6} + \frac{20336}{317275} a^{5} - \frac{20536}{63455} a^{4} - \frac{41872}{317275} a^{3} + \frac{19067}{317275} a^{2} - \frac{11619}{63455} a + \frac{3970}{12691}$, $\frac{1}{239902679140075} a^{26} - \frac{372079181}{239902679140075} a^{25} - \frac{1669085671}{47980535828015} a^{24} - \frac{13096528929}{239902679140075} a^{23} + \frac{4060398581}{47980535828015} a^{22} + \frac{27458434343}{239902679140075} a^{21} - \frac{17153736906}{47980535828015} a^{20} + \frac{37491778448}{239902679140075} a^{19} + \frac{19888439}{31880754703} a^{18} - \frac{3125399970408}{239902679140075} a^{17} - \frac{1009391376933}{239902679140075} a^{16} - \frac{224447750206}{5579132073025} a^{15} - \frac{8921329677032}{239902679140075} a^{14} + \frac{15368694854459}{239902679140075} a^{13} + \frac{2721791037421}{47980535828015} a^{12} + \frac{9808378115153}{34271811305725} a^{11} - \frac{3495246989928}{9596107165603} a^{10} - \frac{65723651702222}{239902679140075} a^{9} + \frac{89431643986}{195838921747} a^{8} + \frac{15412654789428}{239902679140075} a^{7} + \frac{21576088607649}{47980535828015} a^{6} - \frac{1952172573088}{5851284857075} a^{5} - \frac{4439796119548}{239902679140075} a^{4} - \frac{15902389951734}{34271811305725} a^{3} + \frac{107187720306407}{239902679140075} a^{2} - \frac{10224426888601}{47980535828015} a - \frac{3361885110408}{9596107165603}$, $\frac{1}{25974637773290955737442125} a^{27} - \frac{314849447}{633527750568072091157125} a^{26} - \frac{47391558836009694}{120812268712981189476475} a^{25} + \frac{1260363234268205255186}{25974637773290955737442125} a^{24} - \frac{153098375144052646149}{5194927554658191147488425} a^{23} + \frac{7224169299395329353827}{25974637773290955737442125} a^{22} + \frac{129211431934504875307}{5194927554658191147488425} a^{21} - \frac{3414811189977577279004}{25974637773290955737442125} a^{20} - \frac{18689322019588240262}{15145561383843122878975} a^{19} + \frac{52533451903175870390964}{5194927554658191147488425} a^{18} - \frac{21160207840310393390447}{5194927554658191147488425} a^{17} + \frac{367336420210594628003828}{25974637773290955737442125} a^{16} + \frac{36727865296166722183411}{25974637773290955737442125} a^{15} - \frac{230460392031535652070606}{25974637773290955737442125} a^{14} + \frac{36114056255241525781172}{1038985510931638229497685} a^{13} - \frac{1120963118776252889276569}{25974637773290955737442125} a^{12} - \frac{446591093297662333312507}{1038985510931638229497685} a^{11} - \frac{11652570999457683460235953}{25974637773290955737442125} a^{10} + \frac{113926294469806835449633}{1038985510931638229497685} a^{9} - \frac{4143666150335868965602859}{25974637773290955737442125} a^{8} - \frac{1299272957749327239248924}{5194927554658191147488425} a^{7} - \frac{45645616744665658536731}{140403447423194355337525} a^{6} - \frac{1454747202573961108124507}{5194927554658191147488425} a^{5} - \frac{324277694356358538197931}{3710662539041565105348875} a^{4} + \frac{455529277338756412579327}{5194927554658191147488425} a^{3} - \frac{2278806181125286183753649}{25974637773290955737442125} a^{2} + \frac{422813904623347664778272}{1038985510931638229497685} a + \frac{295641202301014397180982}{1038985510931638229497685}$

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:  $13$
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:  \( 4093870424302.4214 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 4093870424302.4214 \cdot 1}{2\sqrt{20444429365013571404907915184985572799247638801}}\approx 2.13961068156221$ (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

\(\Q(\sqrt{-7}) \), 4.0.22001.1, 7.1.31047965207.1, 14.0.6747833004465577869943.1

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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{14}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ $28$ $28$ $28$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
449Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.449.2t1.a.a$1$ $ 449 $ \(\Q(\sqrt{449}) \) $C_2$ (as 2T1) $1$ $1$
1.3143.2t1.a.a$1$ $ 7 \cdot 449 $ \(\Q(\sqrt{-3143}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3143.4t3.c.a$2$ $ 7 \cdot 449 $ 4.2.1411207.1 $D_{4}$ (as 4T3) $1$ $0$
* 2.3143.7t2.a.b$2$ $ 7 \cdot 449 $ 7.1.31047965207.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3143.14t3.a.a$2$ $ 7 \cdot 449 $ 14.2.432825288429292066229201.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3143.14t3.a.c$2$ $ 7 \cdot 449 $ 14.2.432825288429292066229201.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3143.7t2.a.a$2$ $ 7 \cdot 449 $ 7.1.31047965207.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3143.7t2.a.c$2$ $ 7 \cdot 449 $ 7.1.31047965207.1 $D_{7}$ (as 7T2) $1$ $0$
* 2.3143.14t3.a.b$2$ $ 7 \cdot 449 $ 14.2.432825288429292066229201.1 $D_{14}$ (as 14T3) $1$ $0$
* 2.3143.28t10.a.b$2$ $ 7 \cdot 449 $ 28.0.20444429365013571404907915184985572799247638801.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3143.28t10.a.c$2$ $ 7 \cdot 449 $ 28.0.20444429365013571404907915184985572799247638801.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3143.28t10.a.a$2$ $ 7 \cdot 449 $ 28.0.20444429365013571404907915184985572799247638801.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3143.28t10.a.d$2$ $ 7 \cdot 449 $ 28.0.20444429365013571404907915184985572799247638801.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3143.28t10.a.e$2$ $ 7 \cdot 449 $ 28.0.20444429365013571404907915184985572799247638801.1 $D_{28}$ (as 28T10) $1$ $0$
* 2.3143.28t10.a.f$2$ $ 7 \cdot 449 $ 28.0.20444429365013571404907915184985572799247638801.1 $D_{28}$ (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 *.