Normalized defining polynomial
\( x^{28} - x^{27} + 14 x^{26} - 11 x^{25} + 115 x^{24} - 79 x^{23} + 617 x^{22} - 353 x^{21} + 2421 x^{20} - 1188 x^{19} + 7015 x^{18} - 2803 x^{17} + 15415 x^{16} - 5107 x^{15} + 25195 x^{14} - 6334 x^{13} + 30532 x^{12} - 6088 x^{11} + 26057 x^{10} - 3239 x^{9} + 15207 x^{8} - 1590 x^{7} + 5362 x^{6} - 70 x^{5} + 1050 x^{4} - 84 x^{3} + 77 x^{2} + 7 x + 1 \)
Invariants
| Degree: | $28$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 14]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(503553375386417026489977159144851919778199649=3^{14}\cdot 29^{26}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $39.49$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(87=3\cdot 29\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{87}(64,·)$, $\chi_{87}(1,·)$, $\chi_{87}(86,·)$, $\chi_{87}(67,·)$, $\chi_{87}(4,·)$, $\chi_{87}(5,·)$, $\chi_{87}(7,·)$, $\chi_{87}(74,·)$, $\chi_{87}(13,·)$, $\chi_{87}(16,·)$, $\chi_{87}(82,·)$, $\chi_{87}(83,·)$, $\chi_{87}(20,·)$, $\chi_{87}(22,·)$, $\chi_{87}(23,·)$, $\chi_{87}(25,·)$, $\chi_{87}(28,·)$, $\chi_{87}(80,·)$, $\chi_{87}(34,·)$, $\chi_{87}(35,·)$, $\chi_{87}(38,·)$, $\chi_{87}(71,·)$, $\chi_{87}(49,·)$, $\chi_{87}(52,·)$, $\chi_{87}(53,·)$, $\chi_{87}(59,·)$, $\chi_{87}(62,·)$, $\chi_{87}(65,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{17} a^{26} + \frac{6}{17} a^{25} + \frac{8}{17} a^{24} - \frac{5}{17} a^{23} + \frac{2}{17} a^{22} + \frac{5}{17} a^{21} - \frac{5}{17} a^{20} + \frac{1}{17} a^{19} - \frac{1}{17} a^{18} - \frac{2}{17} a^{17} - \frac{6}{17} a^{16} + \frac{5}{17} a^{15} - \frac{4}{17} a^{14} - \frac{3}{17} a^{13} + \frac{2}{17} a^{12} - \frac{5}{17} a^{11} + \frac{5}{17} a^{10} + \frac{1}{17} a^{9} + \frac{1}{17} a^{8} + \frac{1}{17} a^{7} + \frac{2}{17} a^{6} + \frac{8}{17} a^{5} + \frac{1}{17} a^{4} - \frac{5}{17} a^{3} - \frac{2}{17} a^{2} + \frac{6}{17} a - \frac{6}{17}$, $\frac{1}{3895907347685506947339847625632831500841} a^{27} + \frac{94974196297919567674530297273816680935}{3895907347685506947339847625632831500841} a^{26} - \frac{1757113464741665998113530038316256390903}{3895907347685506947339847625632831500841} a^{25} + \frac{24578954983254063362995042355463027681}{3895907347685506947339847625632831500841} a^{24} - \frac{307144181563506270526384407919579491408}{3895907347685506947339847625632831500841} a^{23} + \frac{424237376936331706001988848616728841772}{3895907347685506947339847625632831500841} a^{22} - \frac{1088401692781031861288783548562989117133}{3895907347685506947339847625632831500841} a^{21} - \frac{1802550801714419401257485572078452070133}{3895907347685506947339847625632831500841} a^{20} + \frac{687727648764306980147201738082910313485}{3895907347685506947339847625632831500841} a^{19} - \frac{1535578479753183437920119565859459594718}{3895907347685506947339847625632831500841} a^{18} + \frac{155154103626356511406811380802035561606}{3895907347685506947339847625632831500841} a^{17} - \frac{1778103150074836056109172676602648103048}{3895907347685506947339847625632831500841} a^{16} - \frac{1538325354226302230179671647617129862406}{3895907347685506947339847625632831500841} a^{15} - \frac{1818183532261965231037310205997831490806}{3895907347685506947339847625632831500841} a^{14} - \frac{1572084019506622842671319022828475296847}{3895907347685506947339847625632831500841} a^{13} - \frac{516485973011675242874436686084477689569}{3895907347685506947339847625632831500841} a^{12} + \frac{1725074057080875135854069638030427506029}{3895907347685506947339847625632831500841} a^{11} - \frac{36375549898676541406401752818992323114}{3895907347685506947339847625632831500841} a^{10} + \frac{1406999444225789946382404844468854014208}{3895907347685506947339847625632831500841} a^{9} + \frac{752043549368866031985994752216068354425}{3895907347685506947339847625632831500841} a^{8} - \frac{296748472292101331385986882560244289680}{3895907347685506947339847625632831500841} a^{7} - \frac{83384469760869518667619088021452574201}{229171020452088643961167507390166558873} a^{6} + \frac{810715063244245183211705340945781853939}{3895907347685506947339847625632831500841} a^{5} + \frac{250897107727719820246953299519325099740}{3895907347685506947339847625632831500841} a^{4} + \frac{912927810758554317674830797681254603670}{3895907347685506947339847625632831500841} a^{3} - \frac{989103005516448858452844619781883190451}{3895907347685506947339847625632831500841} a^{2} + \frac{1427458978851209173918301073010047240985}{3895907347685506947339847625632831500841} a - \frac{687849409610446700814666501434686311070}{3895907347685506947339847625632831500841}$
Class group and class number
$C_{4}\times C_{4}\times C_{12}$, which has order $192$ (assuming GRH)
Unit group
| Rank: | $13$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{459718160048198964470148224851155793007}{3895907347685506947339847625632831500841} a^{27} + \frac{491963445984988705273217591819646684565}{3895907347685506947339847625632831500841} a^{26} - \frac{6448629807164339806286652834135729651835}{3895907347685506947339847625632831500841} a^{25} + \frac{5489048472554384718748952494620363866639}{3895907347685506947339847625632831500841} a^{24} - \frac{52951072199703845948981282554345438827087}{3895907347685506947339847625632831500841} a^{23} + \frac{39819307479811215431616417681920535084871}{3895907347685506947339847625632831500841} a^{22} - \frac{283987576946209170566480179195940845846935}{3895907347685506947339847625632831500841} a^{21} + \frac{180713865062290080350196112311444834870074}{3895907347685506947339847625632831500841} a^{20} - \frac{1112673865127612497690665577819613726783806}{3895907347685506947339847625632831500841} a^{19} + \frac{36344069922386707864807676303648846199636}{229171020452088643961167507390166558873} a^{18} - \frac{3217951691774023788210898570050862866706706}{3895907347685506947339847625632831500841} a^{17} + \frac{87876390160312095317718859986505853860340}{229171020452088643961167507390166558873} a^{16} - \frac{7047879163309816292391973949049811453306036}{3895907347685506947339847625632831500841} a^{15} + \frac{2797655820669082469076957101458329888476089}{3895907347685506947339847625632831500841} a^{14} - \frac{11468990009854706273972141903057069947674868}{3895907347685506947339847625632831500841} a^{13} + \frac{3642273987431523561845329655519556538625641}{3895907347685506947339847625632831500841} a^{12} - \frac{13797351382373102873603284632167040793349857}{3895907347685506947339847625632831500841} a^{11} + \frac{3691710695278352959379085590040921104499799}{3895907347685506947339847625632831500841} a^{10} - \frac{11655967246241058805749943256068328625815037}{3895907347685506947339847625632831500841} a^{9} + \frac{2250968703208521239456502251110489534766072}{3895907347685506947339847625632831500841} a^{8} - \frac{392647765542212833797689295166405365735365}{229171020452088643961167507390166558873} a^{7} + \frac{1198090561014405273787508711780039774147370}{3895907347685506947339847625632831500841} a^{6} - \frac{2288207023706649952564153692852833900212139}{3895907347685506947339847625632831500841} a^{5} + \frac{197340796559132081521154403951415280347726}{3895907347685506947339847625632831500841} a^{4} - \frac{415651585504784233687805415050104789263488}{3895907347685506947339847625632831500841} a^{3} + \frac{85650487097946204886580171571688541211264}{3895907347685506947339847625632831500841} a^{2} - \frac{27260512845676607370643210950855091389682}{3895907347685506947339847625632831500841} a + \frac{1495905399110794953147810115657088139776}{3895907347685506947339847625632831500841} \) (order $6$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| 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) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 487075979.1876791 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{14}$ (as 28T2):
| 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
| \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{29})\), 7.7.594823321.1, \(\Q(\zeta_{29})^+\), 14.0.22439994995240462987343.1, 14.0.773792930870360792667.1 |
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 | ${\href{/LocalNumberField/5.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/7.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/13.7.0.1}{7} }^{4}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ | R | ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ | ${\href{/LocalNumberField/43.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/47.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 29 | Data not computed | ||||||