↑

Properties

Label 28.0.186...125.1
Degree $28$
Signature $[0, 14]$
Discriminant $1.863\times 10^{49}$
Root discriminant \(57.50\)
Ramified primes $5,29$
Class number $24224$ (GRH)
Class group [2, 2, 2, 2, 1514] (GRH)
Galois group $C_{28}$ (as 28T1)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 30*x^26 - 30*x^25 + 407*x^24 - 407*x^23 + 3307*x^22 - 3307*x^21 + 17981*x^20 - 17981*x^19 + 69340*x^18 - 69340*x^17 + 196621*x^16 - 196621*x^15 + 421429*x^14 - 421429*x^13 + 702439*x^12 - 702439*x^11 + 945981*x^10 - 945981*x^9 + 1086979*x^8 - 1086979*x^7 + 1138251*x^6 - 1138251*x^5 + 1148807*x^4 - 1148807*x^3 + 1149822*x^2 - 1149822*x + 1149851)
 
gp: K = bnfinit(y^28 - y^27 + 30*y^26 - 30*y^25 + 407*y^24 - 407*y^23 + 3307*y^22 - 3307*y^21 + 17981*y^20 - 17981*y^19 + 69340*y^18 - 69340*y^17 + 196621*y^16 - 196621*y^15 + 421429*y^14 - 421429*y^13 + 702439*y^12 - 702439*y^11 + 945981*y^10 - 945981*y^9 + 1086979*y^8 - 1086979*y^7 + 1138251*y^6 - 1138251*y^5 + 1148807*y^4 - 1148807*y^3 + 1149822*y^2 - 1149822*y + 1149851, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - x^27 + 30*x^26 - 30*x^25 + 407*x^24 - 407*x^23 + 3307*x^22 - 3307*x^21 + 17981*x^20 - 17981*x^19 + 69340*x^18 - 69340*x^17 + 196621*x^16 - 196621*x^15 + 421429*x^14 - 421429*x^13 + 702439*x^12 - 702439*x^11 + 945981*x^10 - 945981*x^9 + 1086979*x^8 - 1086979*x^7 + 1138251*x^6 - 1138251*x^5 + 1148807*x^4 - 1148807*x^3 + 1149822*x^2 - 1149822*x + 1149851);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - x^27 + 30*x^26 - 30*x^25 + 407*x^24 - 407*x^23 + 3307*x^22 - 3307*x^21 + 17981*x^20 - 17981*x^19 + 69340*x^18 - 69340*x^17 + 196621*x^16 - 196621*x^15 + 421429*x^14 - 421429*x^13 + 702439*x^12 - 702439*x^11 + 945981*x^10 - 945981*x^9 + 1086979*x^8 - 1086979*x^7 + 1138251*x^6 - 1138251*x^5 + 1148807*x^4 - 1148807*x^3 + 1149822*x^2 - 1149822*x + 1149851)
 

\( x^{28} - x^{27} + 30 x^{26} - 30 x^{25} + 407 x^{24} - 407 x^{23} + 3307 x^{22} - 3307 x^{21} + \cdots + 1149851 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(18634854406558377293367533932433545897882080078125\) \(\medspace = 5^{14}\cdot 29^{27}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(57.50\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $5^{1/2}29^{27/28}\approx 57.498233227148596$
Ramified primes:   \(5\), \(29\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{29}) \)
$\card{ \Gal(K/\Q) }$:  $28$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(145=5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{145}(1,·)$, $\chi_{145}(69,·)$, $\chi_{145}(6,·)$, $\chi_{145}(71,·)$, $\chi_{145}(136,·)$, $\chi_{145}(141,·)$, $\chi_{145}(14,·)$, $\chi_{145}(79,·)$, $\chi_{145}(16,·)$, $\chi_{145}(81,·)$, $\chi_{145}(19,·)$, $\chi_{145}(84,·)$, $\chi_{145}(86,·)$, $\chi_{145}(89,·)$, $\chi_{145}(91,·)$, $\chi_{145}(96,·)$, $\chi_{145}(99,·)$, $\chi_{145}(36,·)$, $\chi_{145}(134,·)$, $\chi_{145}(39,·)$, $\chi_{145}(104,·)$, $\chi_{145}(44,·)$, $\chi_{145}(111,·)$, $\chi_{145}(114,·)$, $\chi_{145}(51,·)$, $\chi_{145}(119,·)$, $\chi_{145}(121,·)$, $\chi_{145}(124,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{8192}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{514229}a^{15}-\frac{196418}{514229}a^{14}+\frac{15}{514229}a^{13}-\frac{178707}{514229}a^{12}+\frac{90}{514229}a^{11}-\frac{211545}{514229}a^{10}+\frac{275}{514229}a^{9}-\frac{109460}{514229}a^{8}+\frac{450}{514229}a^{7}-\frac{153244}{514229}a^{6}+\frac{378}{514229}a^{5}+\frac{69247}{514229}a^{4}+\frac{140}{514229}a^{3}+\frac{145869}{514229}a^{2}+\frac{15}{514229}a+\frac{121393}{514229}$, $\frac{1}{514229}a^{16}+\frac{16}{514229}a^{14}+\frac{196418}{514229}a^{13}+\frac{104}{514229}a^{12}-\frac{17711}{514229}a^{11}+\frac{352}{514229}a^{10}-\frac{88555}{514229}a^{9}+\frac{660}{514229}a^{8}-\frac{212532}{514229}a^{7}+\frac{672}{514229}a^{6}-\frac{247954}{514229}a^{5}+\frac{336}{514229}a^{4}-\frac{123977}{514229}a^{3}+\frac{64}{514229}a^{2}-\frac{17711}{514229}a+\frac{2}{514229}$, $\frac{1}{514229}a^{17}+\frac{253732}{514229}a^{14}-\frac{136}{514229}a^{13}-\frac{243773}{514229}a^{12}-\frac{1088}{514229}a^{11}+\frac{210791}{514229}a^{10}-\frac{3740}{514229}a^{9}-\frac{3859}{514229}a^{8}-\frac{6528}{514229}a^{7}+\frac{147034}{514229}a^{6}-\frac{5712}{514229}a^{5}-\frac{203471}{514229}a^{4}-\frac{2176}{514229}a^{3}+\frac{219530}{514229}a^{2}-\frac{238}{514229}a+\frac{114628}{514229}$, $\frac{1}{514229}a^{18}-\frac{153}{514229}a^{14}+\frac{64079}{514229}a^{13}-\frac{1326}{514229}a^{12}+\frac{987}{514229}a^{11}-\frac{5049}{514229}a^{10}+\frac{154985}{514229}a^{9}-\frac{10098}{514229}a^{8}+\frac{126472}{514229}a^{7}-\frac{10710}{514229}a^{6}+\frac{46656}{514229}a^{5}-\frac{5508}{514229}a^{4}+\frac{178851}{514229}a^{3}-\frac{1071}{514229}a^{2}-\frac{91749}{514229}a-\frac{34}{514229}$, $\frac{1}{514229}a^{19}-\frac{162593}{514229}a^{14}+\frac{969}{514229}a^{13}-\frac{87047}{514229}a^{12}+\frac{8721}{514229}a^{11}+\frac{185027}{514229}a^{10}+\frac{31977}{514229}a^{9}-\frac{165580}{514229}a^{8}+\frac{58140}{514229}a^{7}+\frac{254858}{514229}a^{6}+\frac{52326}{514229}a^{5}-\frac{25167}{514229}a^{4}+\frac{20349}{514229}a^{3}+\frac{114361}{514229}a^{2}+\frac{2261}{514229}a+\frac{60885}{514229}$, $\frac{1}{514229}a^{20}+\frac{1140}{514229}a^{14}-\frac{219297}{514229}a^{13}+\frac{11115}{514229}a^{12}-\frac{94244}{514229}a^{11}+\frac{45144}{514229}a^{10}-\frac{190428}{514229}a^{9}+\frac{94050}{514229}a^{8}-\frac{113039}{514229}a^{7}+\frac{102600}{514229}a^{6}+\frac{241736}{514229}a^{5}+\frac{53865}{514229}a^{4}+\frac{251305}{514229}a^{3}+\frac{10640}{514229}a^{2}-\frac{71365}{514229}a+\frac{342}{514229}$, $\frac{1}{514229}a^{21}+\frac{7608}{514229}a^{14}-\frac{5985}{514229}a^{13}-\frac{2948}{514229}a^{12}-\frac{57456}{514229}a^{11}-\frac{202529}{514229}a^{10}-\frac{219450}{514229}a^{9}+\frac{227943}{514229}a^{8}+\frac{103829}{514229}a^{7}+\frac{102036}{514229}a^{6}+\frac{137174}{514229}a^{5}-\frac{13238}{514229}a^{4}-\frac{148960}{514229}a^{3}+\frac{248171}{514229}a^{2}-\frac{16758}{514229}a-\frac{60419}{514229}$, $\frac{1}{514229}a^{22}-\frac{7315}{514229}a^{14}-\frac{117068}{514229}a^{13}-\frac{76076}{514229}a^{12}+\frac{141209}{514229}a^{11}+\frac{192369}{514229}a^{10}+\frac{192659}{514229}a^{9}-\frac{175471}{514229}a^{8}-\frac{236190}{514229}a^{7}-\frac{253846}{514229}a^{6}+\frac{196312}{514229}a^{5}+\frac{104589}{514229}a^{4}+\frac{211509}{514229}a^{3}-\frac{81928}{514229}a^{2}-\frac{174539}{514229}a-\frac{2660}{514229}$, $\frac{1}{514229}a^{23}-\frac{158912}{514229}a^{14}+\frac{33649}{514229}a^{13}+\frac{69622}{514229}a^{12}-\frac{177739}{514229}a^{11}+\frac{56045}{514229}a^{10}-\frac{220762}{514229}a^{9}+\frac{232692}{514229}a^{8}-\frac{47470}{514229}a^{7}+\frac{235672}{514229}a^{6}-\frac{215715}{514229}a^{5}+\frac{237749}{514229}a^{4}-\frac{86286}{514229}a^{3}-\frac{167979}{514229}a^{2}+\frac{107065}{514229}a-\frac{83688}{514229}$, $\frac{1}{514229}a^{24}+\frac{42504}{514229}a^{14}-\frac{117843}{514229}a^{13}-\frac{53769}{514229}a^{12}-\frac{40287}{514229}a^{11}-\frac{53156}{514229}a^{10}+\frac{224027}{514229}a^{9}-\frac{244836}{514229}a^{8}-\frac{245988}{514229}a^{7}-\frac{183490}{514229}a^{6}+\frac{141692}{514229}a^{5}+\frac{106607}{514229}a^{4}-\frac{32146}{514229}a^{3}+\frac{26731}{514229}a^{2}+\frac{243076}{514229}a+\frac{17710}{514229}$, $\frac{1}{514229}a^{25}-\frac{74986}{514229}a^{14}-\frac{177100}{514229}a^{13}+\frac{45482}{514229}a^{12}+\frac{235316}{514229}a^{11}-\frac{75587}{514229}a^{10}-\frac{106169}{514229}a^{9}+\frac{12089}{514229}a^{8}+\frac{230412}{514229}a^{7}-\frac{114075}{514229}a^{6}-\frac{18806}{514229}a^{5}+\frac{140162}{514229}a^{4}+\frac{246919}{514229}a^{3}-\frac{228076}{514229}a^{2}-\frac{105621}{514229}a+\frac{85714}{514229}$, $\frac{1}{514229}a^{26}-\frac{230230}{514229}a^{14}+\frac{141814}{514229}a^{13}+\frac{5725}{514229}a^{12}-\frac{11824}{514229}a^{11}-\frac{83347}{514229}a^{10}+\frac{64079}{514229}a^{9}-\frac{128079}{514229}a^{8}+\frac{204740}{514229}a^{7}-\frac{212156}{514229}a^{6}+\frac{202275}{514229}a^{5}+\frac{118019}{514229}a^{4}-\frac{14616}{514229}a^{3}-\frac{137846}{514229}a^{2}+\frac{182046}{514229}a-\frac{106260}{514229}$, $\frac{1}{514229}a^{27}+\frac{123934}{514229}a^{14}-\frac{140428}{514229}a^{13}+\frac{252085}{514229}a^{12}+\frac{68193}{514229}a^{11}+\frac{230006}{514229}a^{10}-\frac{64996}{514229}a^{9}+\frac{49543}{514229}a^{8}+\frac{31315}{514229}a^{7}+\frac{87845}{514229}a^{6}+\frac{240258}{514229}a^{5}+\frac{80507}{514229}a^{4}+\frac{212156}{514229}a^{3}-\frac{179845}{514229}a^{2}-\frac{252413}{514229}a-\frac{35760}{514229}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1514}$, which has order $24224$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{89}{514229}a^{18}+\frac{1602}{514229}a^{16}+\frac{12015}{514229}a^{14}+\frac{48594}{514229}a^{12}-\frac{2584}{514229}a^{11}+\frac{114543}{514229}a^{10}-\frac{28424}{514229}a^{9}+\frac{158598}{514229}a^{8}-\frac{113696}{514229}a^{7}+\frac{123354}{514229}a^{6}-\frac{198968}{514229}a^{5}+\frac{48060}{514229}a^{4}-\frac{142120}{514229}a^{3}+\frac{7209}{514229}a^{2}-\frac{28424}{514229}a+\frac{178}{514229}$, $\frac{13}{514229}a^{22}+\frac{286}{514229}a^{20}+\frac{2717}{514229}a^{18}+\frac{14586}{514229}a^{16}+\frac{48620}{514229}a^{14}+\frac{104104}{514229}a^{12}+\frac{143143}{514229}a^{10}+\frac{122694}{514229}a^{8}-\frac{17711}{514229}a^{7}+\frac{61347}{514229}a^{6}-\frac{123977}{514229}a^{5}+\frac{15730}{514229}a^{4}-\frac{247954}{514229}a^{3}+\frac{1573}{514229}a^{2}-\frac{123977}{514229}a+\frac{26}{514229}$, $\frac{8}{514229}a^{23}+\frac{184}{514229}a^{21}+\frac{1840}{514229}a^{19}+\frac{10488}{514229}a^{17}+\frac{37536}{514229}a^{15}+\frac{87584}{514229}a^{13}+\frac{133952}{514229}a^{11}+\frac{131560}{514229}a^{9}+\frac{78936}{514229}a^{7}-\frac{28657}{514229}a^{6}+\frac{26312}{514229}a^{5}-\frac{171942}{514229}a^{4}+\frac{4048}{514229}a^{3}-\frac{257913}{514229}a^{2}+\frac{184}{514229}a-\frac{57314}{514229}$, $\frac{5}{514229}a^{24}+\frac{120}{514229}a^{22}+\frac{1260}{514229}a^{20}+\frac{7600}{514229}a^{18}+\frac{29070}{514229}a^{16}+\frac{73440}{514229}a^{14}+\frac{123760}{514229}a^{12}+\frac{137280}{514229}a^{10}+\frac{96525}{514229}a^{8}+\frac{40040}{514229}a^{6}-\frac{46368}{514229}a^{5}+\frac{8580}{514229}a^{4}-\frac{231840}{514229}a^{3}+\frac{720}{514229}a^{2}-\frac{231840}{514229}a+\frac{514239}{514229}$, $\frac{8}{514229}a^{23}-\frac{13}{514229}a^{22}+\frac{205}{514229}a^{21}-\frac{286}{514229}a^{20}+\frac{2281}{514229}a^{19}-\frac{2717}{514229}a^{18}+\frac{14457}{514229}a^{17}-\frac{14586}{514229}a^{16}+\frac{57528}{514229}a^{15}-\frac{48620}{514229}a^{14}+\frac{149324}{514229}a^{13}-\frac{104104}{514229}a^{12}+\frac{254345}{514229}a^{11}-\frac{143143}{514229}a^{10}+\frac{278707}{514229}a^{9}-\frac{133640}{514229}a^{8}+\frac{204755}{514229}a^{7}-\frac{177572}{514229}a^{6}+\frac{193948}{514229}a^{5}-\frac{406592}{514229}a^{4}+\frac{260087}{514229}a^{3}-\frac{434622}{514229}a^{2}+\frac{124602}{514229}a-\frac{79232}{514229}$, $\frac{377}{514229}a^{15}-\frac{610}{514229}a^{14}+\frac{5655}{514229}a^{13}-\frac{8540}{514229}a^{12}+\frac{33930}{514229}a^{11}-\frac{46970}{514229}a^{10}+\frac{103675}{514229}a^{9}-\frac{128100}{514229}a^{8}+\frac{169650}{514229}a^{7}-\frac{179340}{514229}a^{6}+\frac{142506}{514229}a^{5}-\frac{119560}{514229}a^{4}+\frac{52780}{514229}a^{3}-\frac{29890}{514229}a^{2}+\frac{5655}{514229}a-\frac{515449}{514229}$, $\frac{13}{514229}a^{22}+\frac{286}{514229}a^{20}+\frac{2717}{514229}a^{18}+\frac{14586}{514229}a^{16}+\frac{48620}{514229}a^{14}+\frac{104104}{514229}a^{12}+\frac{143143}{514229}a^{10}+\frac{122694}{514229}a^{8}-\frac{17711}{514229}a^{7}+\frac{61347}{514229}a^{6}-\frac{123977}{514229}a^{5}+\frac{15730}{514229}a^{4}-\frac{247954}{514229}a^{3}+\frac{1573}{514229}a^{2}-\frac{123977}{514229}a+\frac{514255}{514229}$, $\frac{3}{514229}a^{25}+\frac{75}{514229}a^{23}+\frac{846}{514229}a^{21}+\frac{5691}{514229}a^{19}+\frac{25488}{514229}a^{17}-\frac{233}{514229}a^{16}+\frac{80580}{514229}a^{15}-\frac{3728}{514229}a^{14}+\frac{186963}{514229}a^{13}-\frac{25829}{514229}a^{12}+\frac{329472}{514229}a^{11}-\frac{101180}{514229}a^{10}+\frac{453192}{514229}a^{9}-\frac{250964}{514229}a^{8}+\frac{477273}{514229}a^{7}-\frac{423008}{514229}a^{6}+\frac{341124}{514229}a^{5}-\frac{539918}{514229}a^{4}+\frac{129228}{514229}a^{3}-\frac{547640}{514229}a^{2}+\frac{15795}{514229}a-\frac{689831}{514229}$, $\frac{377}{514229}a^{15}-\frac{610}{514229}a^{14}+\frac{5655}{514229}a^{13}-\frac{8540}{514229}a^{12}+\frac{33930}{514229}a^{11}-\frac{46970}{514229}a^{10}+\frac{103675}{514229}a^{9}-\frac{128100}{514229}a^{8}+\frac{169650}{514229}a^{7}-\frac{179340}{514229}a^{6}+\frac{142506}{514229}a^{5}-\frac{119560}{514229}a^{4}+\frac{52780}{514229}a^{3}-\frac{29890}{514229}a^{2}+\frac{5655}{514229}a-\frac{1220}{514229}$, $\frac{34}{514229}a^{20}+\frac{680}{514229}a^{18}+\frac{5780}{514229}a^{16}+\frac{27200}{514229}a^{14}+\frac{77350}{514229}a^{12}+\frac{136136}{514229}a^{10}-\frac{6765}{514229}a^{9}+\frac{145860}{514229}a^{8}-\frac{60885}{514229}a^{7}+\frac{89760}{514229}a^{6}-\frac{182655}{514229}a^{5}+\frac{28050}{514229}a^{4}-\frac{202950}{514229}a^{3}+\frac{3400}{514229}a^{2}-\frac{60885}{514229}a+\frac{68}{514229}$, $\frac{1}{514229}a^{27}-\frac{2}{514229}a^{26}+\frac{30}{514229}a^{25}-\frac{57}{514229}a^{24}+\frac{407}{514229}a^{23}-\frac{731}{514229}a^{22}+\frac{3307}{514229}a^{21}-\frac{5584}{514229}a^{20}+\frac{17926}{514229}a^{19}-\frac{28376}{514229}a^{18}+\frac{68295}{514229}a^{17}-\frac{101659}{514229}a^{16}+\frac{188261}{514229}a^{15}-\frac{266389}{514229}a^{14}+\frac{384854}{514229}a^{13}-\frac{526081}{514229}a^{12}+\frac{607344}{514229}a^{11}-\frac{805664}{514229}a^{10}+\frac{796546}{514229}a^{9}-\frac{977101}{514229}a^{8}+\frac{949039}{514229}a^{7}-\frac{971532}{514229}a^{6}+\frac{1069281}{514229}a^{5}-\frac{936572}{514229}a^{4}+\frac{1133132}{514229}a^{3}-\frac{1045101}{514229}a^{2}+\frac{830966}{514229}a-\frac{1141487}{514229}$, $\frac{144}{514229}a^{17}+\frac{2448}{514229}a^{15}+\frac{17136}{514229}a^{13}-\frac{1597}{514229}a^{12}+\frac{63648}{514229}a^{11}-\frac{19164}{514229}a^{10}+\frac{134640}{514229}a^{9}-\frac{86238}{514229}a^{8}+\frac{161568}{514229}a^{7}-\frac{178864}{514229}a^{6}+\frac{102816}{514229}a^{5}-\frac{167685}{514229}a^{4}+\frac{29376}{514229}a^{3}-\frac{57492}{514229}a^{2}+\frac{2448}{514229}a-\frac{3194}{514229}$, $\frac{2}{514229}a^{26}+\frac{52}{514229}a^{24}+\frac{598}{514229}a^{22}+\frac{4004}{514229}a^{20}+\frac{17290}{514229}a^{18}+\frac{50388}{514229}a^{16}+\frac{100776}{514229}a^{14}+\frac{137904}{514229}a^{12}+\frac{126412}{514229}a^{10}+\frac{74360}{514229}a^{8}+\frac{26026}{514229}a^{6}+\frac{4732}{514229}a^{4}-\frac{121393}{514229}a^{3}+\frac{338}{514229}a^{2}-\frac{364179}{514229}a+\frac{4}{514229}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 487075979.1876791 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 487075979.1876791 \cdot 24224}{2\cdot\sqrt{18634854406558377293367533932433545897882080078125}}\cr\approx \mathstrut & 0.204252663933886 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 + 30*x^26 - 30*x^25 + 407*x^24 - 407*x^23 + 3307*x^22 - 3307*x^21 + 17981*x^20 - 17981*x^19 + 69340*x^18 - 69340*x^17 + 196621*x^16 - 196621*x^15 + 421429*x^14 - 421429*x^13 + 702439*x^12 - 702439*x^11 + 945981*x^10 - 945981*x^9 + 1086979*x^8 - 1086979*x^7 + 1138251*x^6 - 1138251*x^5 + 1148807*x^4 - 1148807*x^3 + 1149822*x^2 - 1149822*x + 1149851)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^28 - x^27 + 30*x^26 - 30*x^25 + 407*x^24 - 407*x^23 + 3307*x^22 - 3307*x^21 + 17981*x^20 - 17981*x^19 + 69340*x^18 - 69340*x^17 + 196621*x^16 - 196621*x^15 + 421429*x^14 - 421429*x^13 + 702439*x^12 - 702439*x^11 + 945981*x^10 - 945981*x^9 + 1086979*x^8 - 1086979*x^7 + 1138251*x^6 - 1138251*x^5 + 1148807*x^4 - 1148807*x^3 + 1149822*x^2 - 1149822*x + 1149851, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^28 - x^27 + 30*x^26 - 30*x^25 + 407*x^24 - 407*x^23 + 3307*x^22 - 3307*x^21 + 17981*x^20 - 17981*x^19 + 69340*x^18 - 69340*x^17 + 196621*x^16 - 196621*x^15 + 421429*x^14 - 421429*x^13 + 702439*x^12 - 702439*x^11 + 945981*x^10 - 945981*x^9 + 1086979*x^8 - 1086979*x^7 + 1138251*x^6 - 1138251*x^5 + 1148807*x^4 - 1148807*x^3 + 1149822*x^2 - 1149822*x + 1149851);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - x^27 + 30*x^26 - 30*x^25 + 407*x^24 - 407*x^23 + 3307*x^22 - 3307*x^21 + 17981*x^20 - 17981*x^19 + 69340*x^18 - 69340*x^17 + 196621*x^16 - 196621*x^15 + 421429*x^14 - 421429*x^13 + 702439*x^12 - 702439*x^11 + 945981*x^10 - 945981*x^9 + 1086979*x^8 - 1086979*x^7 + 1138251*x^6 - 1138251*x^5 + 1148807*x^4 - 1148807*x^3 + 1149822*x^2 - 1149822*x + 1149851);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 28
The 28 conjugacy class representatives for $C_{28}$
Character table for $C_{28}$

Intermediate fields

\(\Q(\sqrt{29}) \), 4.0.609725.2, 7.7.594823321.1, \(\Q(\zeta_{29})^+\)

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

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 ${\href{/padicField/7.14.0.1}{14} }^{2}$ $28$ ${\href{/padicField/13.7.0.1}{7} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/padicField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/padicField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/padicField/53.14.0.1}{14} }^{2}$ ${\href{/padicField/59.1.0.1}{1} }^{28}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display 5.14.7.2$x^{14} + 46875 x^{2} - 234375$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5.14.7.2$x^{14} + 46875 x^{2} - 234375$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
\(29\) Copy content Toggle raw display Deg $28$$28$$1$$27$