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Properties

Label 28.0.596...125.1
Degree $28$
Signature $[0, 14]$
Discriminant $5.969\times 10^{49}$
Root discriminant $59.94$
Ramified primes $5, 29$
Class number $3368$ (GRH)
Class group $[2, 2, 842]$ (GRH)
Galois group $C_{28}$ (as 28T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 13*x^26 - 18*x^25 + 139*x^24 + 6*x^23 + 1195*x^22 + 162*x^21 + 10698*x^20 - 2678*x^19 + 31811*x^18 - 10412*x^17 + 82335*x^16 - 59127*x^15 + 217167*x^14 - 143196*x^13 + 464676*x^12 - 285512*x^11 + 357781*x^10 - 176521*x^9 + 256648*x^8 + 112508*x^7 + 36201*x^6 + 9801*x^5 + 2858*x^4 + 449*x^3 + 67*x^2 + 9*x + 1)
 
gp: K = bnfinit(x^28 - x^27 + 13*x^26 - 18*x^25 + 139*x^24 + 6*x^23 + 1195*x^22 + 162*x^21 + 10698*x^20 - 2678*x^19 + 31811*x^18 - 10412*x^17 + 82335*x^16 - 59127*x^15 + 217167*x^14 - 143196*x^13 + 464676*x^12 - 285512*x^11 + 357781*x^10 - 176521*x^9 + 256648*x^8 + 112508*x^7 + 36201*x^6 + 9801*x^5 + 2858*x^4 + 449*x^3 + 67*x^2 + 9*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 9, 67, 449, 2858, 9801, 36201, 112508, 256648, -176521, 357781, -285512, 464676, -143196, 217167, -59127, 82335, -10412, 31811, -2678, 10698, 162, 1195, 6, 139, -18, 13, -1, 1]);
 

\( x^{28} - x^{27} + 13 x^{26} - 18 x^{25} + 139 x^{24} + 6 x^{23} + 1195 x^{22} + 162 x^{21} + 10698 x^{20} - 2678 x^{19} + 31811 x^{18} - 10412 x^{17} + 82335 x^{16} - 59127 x^{15} + 217167 x^{14} - 143196 x^{13} + 464676 x^{12} - 285512 x^{11} + 357781 x^{10} - 176521 x^{9} + 256648 x^{8} + 112508 x^{7} + 36201 x^{6} + 9801 x^{5} + 2858 x^{4} + 449 x^{3} + 67 x^{2} + 9 x + 1 \)

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:  \(59692812354437378574125162510614243030548095703125\)\(\medspace = 5^{21}\cdot 29^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $59.94$
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:  $5, 29$
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:  \(145=5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{145}(1,·)$, $\chi_{145}(132,·)$, $\chi_{145}(7,·)$, $\chi_{145}(136,·)$, $\chi_{145}(74,·)$, $\chi_{145}(139,·)$, $\chi_{145}(141,·)$, $\chi_{145}(78,·)$, $\chi_{145}(16,·)$, $\chi_{145}(81,·)$, $\chi_{145}(82,·)$, $\chi_{145}(83,·)$, $\chi_{145}(23,·)$, $\chi_{145}(24,·)$, $\chi_{145}(94,·)$, $\chi_{145}(123,·)$, $\chi_{145}(36,·)$, $\chi_{145}(103,·)$, $\chi_{145}(107,·)$, $\chi_{145}(111,·)$, $\chi_{145}(112,·)$, $\chi_{145}(49,·)$, $\chi_{145}(52,·)$, $\chi_{145}(53,·)$, $\chi_{145}(54,·)$, $\chi_{145}(88,·)$, $\chi_{145}(59,·)$, $\chi_{145}(117,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{41} a^{18} + \frac{17}{41} a^{17} + \frac{18}{41} a^{16} + \frac{4}{41} a^{15} - \frac{13}{41} a^{14} + \frac{13}{41} a^{13} + \frac{13}{41} a^{12} + \frac{19}{41} a^{11} - \frac{2}{41} a^{10} - \frac{17}{41} a^{9} - \frac{11}{41} a^{8} - \frac{8}{41} a^{7} + \frac{16}{41} a^{6} - \frac{20}{41} a^{5} - \frac{2}{41} a^{4} - \frac{13}{41} a^{3} - \frac{7}{41} a^{2} + \frac{1}{41} a - \frac{13}{41}$, $\frac{1}{41} a^{19} + \frac{16}{41} a^{17} - \frac{15}{41} a^{16} + \frac{1}{41} a^{15} - \frac{12}{41} a^{14} - \frac{3}{41} a^{13} + \frac{3}{41} a^{12} + \frac{3}{41} a^{11} + \frac{17}{41} a^{10} - \frac{9}{41} a^{9} + \frac{15}{41} a^{8} - \frac{12}{41} a^{7} - \frac{5}{41} a^{6} + \frac{10}{41} a^{5} - \frac{20}{41} a^{4} + \frac{9}{41} a^{3} - \frac{3}{41} a^{2} + \frac{11}{41} a + \frac{16}{41}$, $\frac{1}{41} a^{20} + \frac{6}{41} a^{15} - \frac{18}{41} a^{10} + \frac{13}{41} a^{5} + \frac{3}{41}$, $\frac{1}{41} a^{21} + \frac{6}{41} a^{16} - \frac{18}{41} a^{11} + \frac{13}{41} a^{6} + \frac{3}{41} a$, $\frac{1}{41} a^{22} + \frac{6}{41} a^{17} - \frac{18}{41} a^{12} + \frac{13}{41} a^{7} + \frac{3}{41} a^{2}$, $\frac{1}{41} a^{23} - \frac{20}{41} a^{17} + \frac{15}{41} a^{16} + \frac{17}{41} a^{15} - \frac{4}{41} a^{14} - \frac{14}{41} a^{13} + \frac{4}{41} a^{12} + \frac{9}{41} a^{11} + \frac{12}{41} a^{10} + \frac{20}{41} a^{9} - \frac{3}{41} a^{8} + \frac{7}{41} a^{7} - \frac{14}{41} a^{6} - \frac{3}{41} a^{5} + \frac{12}{41} a^{4} - \frac{1}{41} a^{3} + \frac{1}{41} a^{2} - \frac{6}{41} a - \frac{4}{41}$, $\frac{1}{697} a^{24} + \frac{2}{697} a^{23} - \frac{7}{697} a^{22} + \frac{4}{697} a^{21} - \frac{7}{697} a^{19} - \frac{4}{697} a^{18} + \frac{93}{697} a^{17} - \frac{192}{697} a^{16} + \frac{210}{697} a^{15} + \frac{182}{697} a^{14} + \frac{1}{17} a^{13} - \frac{162}{697} a^{12} + \frac{36}{697} a^{11} + \frac{180}{697} a^{10} + \frac{115}{697} a^{9} - \frac{239}{697} a^{8} - \frac{94}{697} a^{7} + \frac{66}{697} a^{6} - \frac{97}{697} a^{5} + \frac{295}{697} a^{4} - \frac{149}{697} a^{3} + \frac{212}{697} a^{2} - \frac{229}{697} a + \frac{3}{17}$, $\frac{1}{7090839830122299536497} a^{25} - \frac{4868634166802997472}{7090839830122299536497} a^{24} - \frac{43639177814540008192}{7090839830122299536497} a^{23} - \frac{14784432187095961465}{7090839830122299536497} a^{22} - \frac{24124201965568567720}{7090839830122299536497} a^{21} - \frac{41860118079929702280}{7090839830122299536497} a^{20} - \frac{43138260189157573062}{7090839830122299536497} a^{19} + \frac{59677104797356897177}{7090839830122299536497} a^{18} - \frac{2797043695152919982905}{7090839830122299536497} a^{17} - \frac{902711460470222177622}{7090839830122299536497} a^{16} + \frac{1797128177762167604683}{7090839830122299536497} a^{15} + \frac{1731718377972508108859}{7090839830122299536497} a^{14} + \frac{1037486025067334127046}{7090839830122299536497} a^{13} - \frac{2279855663961411084808}{7090839830122299536497} a^{12} + \frac{1985426563204438623375}{7090839830122299536497} a^{11} - \frac{1028177995957344822776}{7090839830122299536497} a^{10} + \frac{228039167825600162451}{7090839830122299536497} a^{9} + \frac{786358035072420640414}{7090839830122299536497} a^{8} + \frac{994760748448112029326}{7090839830122299536497} a^{7} + \frac{2964525256948967550685}{7090839830122299536497} a^{6} + \frac{81982421278732310533}{172947312929812183817} a^{5} + \frac{1011568314517591782981}{7090839830122299536497} a^{4} - \frac{2763577931842279820721}{7090839830122299536497} a^{3} + \frac{115343829192521005545}{417108225301311737441} a^{2} + \frac{3097199355571049520185}{7090839830122299536497} a - \frac{2252620849264754509251}{7090839830122299536497}$, $\frac{1}{290724433035014280996377} a^{26} - \frac{15}{290724433035014280996377} a^{25} - \frac{67743437584379503606}{290724433035014280996377} a^{24} - \frac{2848044891802473221399}{290724433035014280996377} a^{23} + \frac{2035123640789919177994}{290724433035014280996377} a^{22} - \frac{2334664404106304751947}{290724433035014280996377} a^{21} + \frac{2886090701869422877174}{290724433035014280996377} a^{20} - \frac{2101823223174063066704}{290724433035014280996377} a^{19} - \frac{861885122804888363562}{290724433035014280996377} a^{18} + \frac{106428610734794070287905}{290724433035014280996377} a^{17} + \frac{96760685719616172640491}{290724433035014280996377} a^{16} - \frac{848585949345237572086}{7090839830122299536497} a^{15} + \frac{98183475875344261293487}{290724433035014280996377} a^{14} - \frac{61942336099147708271123}{290724433035014280996377} a^{13} + \frac{3795428868157600467654}{290724433035014280996377} a^{12} - \frac{12456094108524939430296}{290724433035014280996377} a^{11} - \frac{70985560501170034112189}{290724433035014280996377} a^{10} + \frac{69772961752750720181695}{290724433035014280996377} a^{9} + \frac{3932719019374216319896}{290724433035014280996377} a^{8} - \frac{66131010358471348376323}{290724433035014280996377} a^{7} + \frac{4960403152956174373934}{290724433035014280996377} a^{6} + \frac{64700231916222234731245}{290724433035014280996377} a^{5} + \frac{11347208218555249588173}{290724433035014280996377} a^{4} + \frac{5037344900690331371523}{290724433035014280996377} a^{3} + \frac{10608095060101514976068}{290724433035014280996377} a^{2} - \frac{127477334672360098061552}{290724433035014280996377} a + \frac{128419245688967520942160}{290724433035014280996377}$, $\frac{1}{290724433035014280996377} a^{27} - \frac{6}{290724433035014280996377} a^{25} + \frac{141612811055950045470}{290724433035014280996377} a^{24} - \frac{932129322978161160624}{290724433035014280996377} a^{23} + \frac{1528485906442013631394}{290724433035014280996377} a^{22} + \frac{1355934406248753757765}{290724433035014280996377} a^{21} - \frac{2013578134111652377818}{290724433035014280996377} a^{20} + \frac{1446115915193919654741}{290724433035014280996377} a^{19} - \frac{2483990970733821042291}{290724433035014280996377} a^{18} - \frac{55127040996782717102623}{290724433035014280996377} a^{17} - \frac{139443778319150895521777}{290724433035014280996377} a^{16} - \frac{71938934600287548231749}{290724433035014280996377} a^{15} - \frac{117346418342151025179067}{290724433035014280996377} a^{14} - \frac{3664427115867057603146}{290724433035014280996377} a^{13} + \frac{22384274851973660559508}{290724433035014280996377} a^{12} + \frac{19106912705655254381030}{290724433035014280996377} a^{11} + \frac{144675199124035737874369}{290724433035014280996377} a^{10} + \frac{46733090934347195784955}{290724433035014280996377} a^{9} + \frac{74035982138100893535925}{290724433035014280996377} a^{8} + \frac{92966525381905181881532}{290724433035014280996377} a^{7} - \frac{31050070224459718964692}{290724433035014280996377} a^{6} - \frac{37420715931147611269361}{290724433035014280996377} a^{5} + \frac{60496999606379241997239}{290724433035014280996377} a^{4} + \frac{41798286877521114881781}{290724433035014280996377} a^{3} - \frac{59964480471345211227150}{290724433035014280996377} a^{2} - \frac{91453937304973970550032}{290724433035014280996377} a - \frac{13874120655021326234744}{290724433035014280996377}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{842}$, which has order $3368$ (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:  \( \frac{733067873224578509887}{17101437237353781235081} a^{27} - \frac{1204368278251242400806}{17101437237353781235081} a^{26} + \frac{10053856931488632215424}{17101437237353781235081} a^{25} - \frac{19374620128389551665140}{17101437237353781235081} a^{24} + \frac{111063700790038484004762}{17101437237353781235081} a^{23} - \frac{62057770212971286490979}{17101437237353781235081} a^{22} + \frac{880497939078027868241430}{17101437237353781235081} a^{21} - \frac{444097711645599426140682}{17101437237353781235081} a^{20} + \frac{7828865083176739134874098}{17101437237353781235081} a^{19} - \frac{6996437147552391289412664}{17101437237353781235081} a^{18} + \frac{25144467198493549097599570}{17101437237353781235081} a^{17} - \frac{22764916831666799023408890}{17101437237353781235081} a^{16} + \frac{66931929242401839038775498}{17101437237353781235081} a^{15} - \frac{82693235080684924154993010}{17101437237353781235081} a^{14} + \frac{191380193619351858536703552}{17101437237353781235081} a^{13} - \frac{210427621688897255732029329}{17101437237353781235081} a^{12} + \frac{419507391414809222330660190}{17101437237353781235081} a^{11} - \frac{435795320373717209395786656}{17101437237353781235081} a^{10} + \frac{421173765837365278283740596}{17101437237353781235081} a^{9} - \frac{312970020797468016287883870}{17101437237353781235081} a^{8} + \frac{289895794526881617856735776}{17101437237353781235081} a^{7} - \frac{47716076271388712024037402}{17101437237353781235081} a^{6} - \frac{13044355730224651933251420}{17101437237353781235081} a^{5} - \frac{3984207355969664371744614}{17101437237353781235081} a^{4} - \frac{628156602865300166959512}{17101437237353781235081} a^{3} - \frac{736046948085049638095607}{17101437237353781235081} a^{2} - \frac{12829140355284973399890}{17101437237353781235081} a - \frac{1466187469175425531416}{17101437237353781235081} \) (order $10$)
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:  \( 34681517373.86067 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{14}\cdot 34681517373.86067 \cdot 3368}{10\sqrt{59692812354437378574125162510614243030548095703125}}\approx 0.225957785535394$ (assuming GRH)

Galois group

$C_{28}$ (as 28T1):

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

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 7.7.594823321.1, 14.14.27641779937927268828125.1

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 $28$ $28$ R $28$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{7}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ $28$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{4}$ $28$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{28}$ $28$ $28$ $28$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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
5Data not computed
$29$29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$
29.14.12.1$x^{14} + 2407 x^{7} + 1839267$$7$$2$$12$$C_{14}$$[\ ]_{7}^{2}$