Properties

Label 28.0.860...281.1
Degree $28$
Signature $[0, 14]$
Discriminant $8.602\times 10^{48}$
Root discriminant $55.93$
Ramified primes $3, 7$
Class number $203$ (GRH)
Class group $[203]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

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 - 56*x^25 - 7*x^24 - 14*x^23 + 1162*x^22 + 286*x^21 + 539*x^20 - 11074*x^19 - 4074*x^18 - 6433*x^17 + 48363*x^16 + 27475*x^15 + 29189*x^14 - 79527*x^13 - 90503*x^12 - 49217*x^11 + 5593*x^10 + 93345*x^9 + 89726*x^8 + 166311*x^7 + 114709*x^6 + 86730*x^5 + 11690*x^4 - 17626*x^3 + 25221*x^2 - 9401*x + 6241)
 
gp: K = bnfinit(x^28 - 56*x^25 - 7*x^24 - 14*x^23 + 1162*x^22 + 286*x^21 + 539*x^20 - 11074*x^19 - 4074*x^18 - 6433*x^17 + 48363*x^16 + 27475*x^15 + 29189*x^14 - 79527*x^13 - 90503*x^12 - 49217*x^11 + 5593*x^10 + 93345*x^9 + 89726*x^8 + 166311*x^7 + 114709*x^6 + 86730*x^5 + 11690*x^4 - 17626*x^3 + 25221*x^2 - 9401*x + 6241, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6241, -9401, 25221, -17626, 11690, 86730, 114709, 166311, 89726, 93345, 5593, -49217, -90503, -79527, 29189, 27475, 48363, -6433, -4074, -11074, 539, 286, 1162, -14, -7, -56, 0, 0, 1]);
 

\( x^{28} - 56 x^{25} - 7 x^{24} - 14 x^{23} + 1162 x^{22} + 286 x^{21} + 539 x^{20} - 11074 x^{19} - 4074 x^{18} - 6433 x^{17} + 48363 x^{16} + 27475 x^{15} + 29189 x^{14} - 79527 x^{13} - 90503 x^{12} - 49217 x^{11} + 5593 x^{10} + 93345 x^{9} + 89726 x^{8} + 166311 x^{7} + 114709 x^{6} + 86730 x^{5} + 11690 x^{4} - 17626 x^{3} + 25221 x^{2} - 9401 x + 6241 \)

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:  \(8602002546566250227508255640254075178338897178281\)\(\medspace = 3^{14}\cdot 7^{50}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $55.93$
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:  $3, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $28$
This field is Galois and abelian over $\Q$.
Conductor:  \(147=3\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{147}(64,·)$, $\chi_{147}(1,·)$, $\chi_{147}(134,·)$, $\chi_{147}(71,·)$, $\chi_{147}(8,·)$, $\chi_{147}(139,·)$, $\chi_{147}(76,·)$, $\chi_{147}(13,·)$, $\chi_{147}(146,·)$, $\chi_{147}(83,·)$, $\chi_{147}(20,·)$, $\chi_{147}(85,·)$, $\chi_{147}(22,·)$, $\chi_{147}(92,·)$, $\chi_{147}(29,·)$, $\chi_{147}(97,·)$, $\chi_{147}(34,·)$, $\chi_{147}(104,·)$, $\chi_{147}(41,·)$, $\chi_{147}(106,·)$, $\chi_{147}(43,·)$, $\chi_{147}(113,·)$, $\chi_{147}(50,·)$, $\chi_{147}(118,·)$, $\chi_{147}(55,·)$, $\chi_{147}(125,·)$, $\chi_{147}(62,·)$, $\chi_{147}(127,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{67} a^{23} + \frac{3}{67} a^{22} + \frac{5}{67} a^{21} - \frac{9}{67} a^{20} + \frac{6}{67} a^{19} + \frac{14}{67} a^{18} - \frac{1}{67} a^{17} - \frac{16}{67} a^{16} + \frac{11}{67} a^{15} - \frac{15}{67} a^{14} + \frac{11}{67} a^{13} + \frac{16}{67} a^{12} + \frac{26}{67} a^{11} - \frac{20}{67} a^{10} - \frac{26}{67} a^{9} + \frac{1}{67} a^{8} - \frac{29}{67} a^{7} - \frac{11}{67} a^{6} + \frac{18}{67} a^{5} + \frac{13}{67} a^{4} + \frac{13}{67} a^{3} - \frac{21}{67} a^{2} + \frac{32}{67} a - \frac{22}{67}$, $\frac{1}{67} a^{24} - \frac{4}{67} a^{22} - \frac{24}{67} a^{21} + \frac{33}{67} a^{20} - \frac{4}{67} a^{19} + \frac{24}{67} a^{18} - \frac{13}{67} a^{17} - \frac{8}{67} a^{16} + \frac{19}{67} a^{15} - \frac{11}{67} a^{14} - \frac{17}{67} a^{13} - \frac{22}{67} a^{12} - \frac{31}{67} a^{11} - \frac{33}{67} a^{10} + \frac{12}{67} a^{9} - \frac{32}{67} a^{8} + \frac{9}{67} a^{7} - \frac{16}{67} a^{6} + \frac{26}{67} a^{5} - \frac{26}{67} a^{4} + \frac{7}{67} a^{3} + \frac{28}{67} a^{2} + \frac{16}{67} a - \frac{1}{67}$, $\frac{1}{24991} a^{25} - \frac{180}{24991} a^{24} + \frac{48}{24991} a^{23} - \frac{8394}{24991} a^{22} - \frac{5504}{24991} a^{21} - \frac{1789}{24991} a^{20} + \frac{7354}{24991} a^{19} - \frac{7357}{24991} a^{18} - \frac{3348}{24991} a^{17} - \frac{8820}{24991} a^{16} - \frac{8219}{24991} a^{15} - \frac{626}{24991} a^{14} - \frac{8785}{24991} a^{13} + \frac{1612}{24991} a^{12} - \frac{5295}{24991} a^{11} - \frac{9895}{24991} a^{10} + \frac{8650}{24991} a^{9} - \frac{6775}{24991} a^{8} - \frac{1067}{24991} a^{7} + \frac{4277}{24991} a^{6} + \frac{6615}{24991} a^{5} + \frac{4559}{24991} a^{4} - \frac{12348}{24991} a^{3} + \frac{10634}{24991} a^{2} - \frac{3428}{24991} a - \frac{2371}{24991}$, $\frac{1}{4018632481368851789} a^{26} - \frac{9164187088120}{4018632481368851789} a^{25} + \frac{9756}{1974289} a^{24} + \frac{15742965953823085}{4018632481368851789} a^{23} + \frac{301231894285664662}{4018632481368851789} a^{22} + \frac{1854086237406800177}{4018632481368851789} a^{21} + \frac{1198119516069590475}{4018632481368851789} a^{20} + \frac{1661002814701538776}{4018632481368851789} a^{19} + \frac{1935970873799544448}{4018632481368851789} a^{18} + \frac{961526220515910162}{4018632481368851789} a^{17} - \frac{1068721411255719431}{4018632481368851789} a^{16} - \frac{149752326454546722}{4018632481368851789} a^{15} + \frac{1826110108905038293}{4018632481368851789} a^{14} + \frac{1062781666359313279}{4018632481368851789} a^{13} + \frac{3293706990172020}{10773813622972793} a^{12} + \frac{795250459292100193}{4018632481368851789} a^{11} + \frac{1469651075958454424}{4018632481368851789} a^{10} - \frac{1234340535840071214}{4018632481368851789} a^{9} + \frac{642378447644951881}{4018632481368851789} a^{8} + \frac{990350867192513289}{4018632481368851789} a^{7} + \frac{351312950846234035}{4018632481368851789} a^{6} - \frac{160676330058184782}{4018632481368851789} a^{5} + \frac{709261909156301780}{4018632481368851789} a^{4} - \frac{1518603145442654385}{4018632481368851789} a^{3} + \frac{1748179704066416138}{4018632481368851789} a^{2} + \frac{147954540001039751}{4018632481368851789} a - \frac{7777273381888533}{50868765586947491}$, $\frac{1}{782264708700309765510040504606662037229469731219867} a^{27} + \frac{46570544129674953548597494147345}{782264708700309765510040504606662037229469731219867} a^{26} - \frac{15571210237899180873231819841615163966596509594}{782264708700309765510040504606662037229469731219867} a^{25} + \frac{5431185816534497959576448086597312658376869693962}{782264708700309765510040504606662037229469731219867} a^{24} + \frac{5345063811545835627305215989836266743339307443857}{782264708700309765510040504606662037229469731219867} a^{23} - \frac{17643520014139582266001129270227195024286840603538}{782264708700309765510040504606662037229469731219867} a^{22} + \frac{334527106746745653492232316276616148518406142932478}{782264708700309765510040504606662037229469731219867} a^{21} + \frac{267747318502634658780543676799492881411436764536136}{782264708700309765510040504606662037229469731219867} a^{20} - \frac{178247683767725129887999732779477146406617534075002}{782264708700309765510040504606662037229469731219867} a^{19} - \frac{118002627939745659131896032085348004978145414135377}{782264708700309765510040504606662037229469731219867} a^{18} - \frac{387721114163107634374880317769466604574837349838871}{782264708700309765510040504606662037229469731219867} a^{17} + \frac{251594806548042183853740916102347961988790687888222}{782264708700309765510040504606662037229469731219867} a^{16} + \frac{315594152559265547115289102358598174676656995485796}{782264708700309765510040504606662037229469731219867} a^{15} + \frac{294223761357691292785032530434396013982190785371200}{782264708700309765510040504606662037229469731219867} a^{14} - \frac{21624304918643621982866880144013875852833789396280}{782264708700309765510040504606662037229469731219867} a^{13} + \frac{112307926374413900417568654118953540781517211079680}{782264708700309765510040504606662037229469731219867} a^{12} - \frac{284673818578287419318440859883003216272235259561001}{782264708700309765510040504606662037229469731219867} a^{11} + \frac{66037714383462223528495968409832649010373499962904}{782264708700309765510040504606662037229469731219867} a^{10} - \frac{25407370584508019756528125550487107001212313026458}{782264708700309765510040504606662037229469731219867} a^{9} + \frac{212997727842107637610137322940990189168835572837086}{782264708700309765510040504606662037229469731219867} a^{8} + \frac{382273517765169514408764544838909694248115665824183}{782264708700309765510040504606662037229469731219867} a^{7} + \frac{254758651519916051192699650667445315722141046465001}{782264708700309765510040504606662037229469731219867} a^{6} - \frac{226342105561449984160812958926182792712703653664380}{782264708700309765510040504606662037229469731219867} a^{5} - \frac{209007935390394731906414941355904987535722446464279}{782264708700309765510040504606662037229469731219867} a^{4} - \frac{319274686229800315414800049874098367085205307485340}{782264708700309765510040504606662037229469731219867} a^{3} - \frac{58196733586223682722059289603256238231454330776084}{782264708700309765510040504606662037229469731219867} a^{2} - \frac{143405751459412164995689368382476853955682039751209}{782264708700309765510040504606662037229469731219867} a + \frac{3435055002879791884600267703063537550803500727000}{9902084920257085639367601324134962496575566217973}$

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

Class group and class number

$C_{203}$, which has order $203$ (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:  \( \frac{19484256624830066084479257549146132}{384313971639714861166486517483616837367} a^{27} + \frac{23328403931461934586061990919318}{5736029427458430763678903246024131901} a^{26} - \frac{5655625859506795731327806991442230}{384313971639714861166486517483616837367} a^{25} - \frac{1085972559700393341024194990764530295}{384313971639714861166486517483616837367} a^{24} - \frac{222036171646406629478318164483563865}{384313971639714861166486517483616837367} a^{23} + \frac{33296530476837137387280139528892527}{384313971639714861166486517483616837367} a^{22} + \frac{22366582829669234472744771410677140515}{384313971639714861166486517483616837367} a^{21} + \frac{7330137531720316711425613221144859322}{384313971639714861166486517483616837367} a^{20} + \frac{4273825358624491215052055672157386632}{384313971639714861166486517483616837367} a^{19} - \frac{210392141229235345885710042539200520084}{384313971639714861166486517483616837367} a^{18} - \frac{96252763176466774627613656742524547413}{384313971639714861166486517483616837367} a^{17} - \frac{65335643796080313174454350346530745104}{384313971639714861166486517483616837367} a^{16} + \frac{895370921593694027819081519052215296443}{384313971639714861166486517483616837367} a^{15} + \frac{7702947804201029881771825302087026560}{4864733818224238748942867309919200473} a^{14} + \frac{292283491654081221832669361459408075834}{384313971639714861166486517483616837367} a^{13} - \frac{1386412506441237599875895846208488191235}{384313971639714861166486517483616837367} a^{12} - \frac{1843369365384312021180612591556002187684}{384313971639714861166486517483616837367} a^{11} - \frac{419737301740435974203965496160546950677}{384313971639714861166486517483616837367} a^{10} + \frac{41108641134642716326708557166932453056}{384313971639714861166486517483616837367} a^{9} + \frac{1551177707596446060803269897516068972624}{384313971639714861166486517483616837367} a^{8} + \frac{1364601089720989340066862228966628860875}{384313971639714861166486517483616837367} a^{7} + \frac{3012337905816300510527454308246005547310}{384313971639714861166486517483616837367} a^{6} + \frac{2522796857281906483060616587693390030566}{384313971639714861166486517483616837367} a^{5} + \frac{1698998269373640968427978378626612129699}{384313971639714861166486517483616837367} a^{4} + \frac{671711984447844853653502508932888447877}{384313971639714861166486517483616837367} a^{3} - \frac{25983582601985266877220218123311081900}{384313971639714861166486517483616837367} a^{2} + \frac{735194648674975386751851126891452695942}{384313971639714861166486517483616837367} a + \frac{1231915933935924231178229402334538943}{4864733818224238748942867309919200473} \) (order $6$)
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:  \( 156859247232.13297 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 156859247232.13297 \cdot 203}{6\sqrt{8602002546566250227508255640254075178338897178281}}\approx 0.270441469841313$ (assuming GRH)

Galois group

$C_2\times C_{14}$ (as 28T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}, \sqrt{-7})\), 7.7.13841287201.1, 14.0.1341068619663964900807.1, 14.0.418988153029298748294987.1, 14.14.2932917071205091238064909.1

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{2}$

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
3Data not computed
$7$7.14.25.59$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$14$$1$$25$$C_{14}$$[2]_{2}$
7.14.25.59$x^{14} - 168 x^{13} + 70 x^{12} - 147 x^{11} + 147 x^{10} - 98 x^{9} + 49 x^{8} + 168 x^{7} - 49 x^{4} - 147 x^{3} - 49 x^{2} + 98 x + 126$$14$$1$$25$$C_{14}$$[2]_{2}$