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Properties

Label 28.28.353...381.1
Degree $28$
Signature $[28, 0]$
Discriminant $3.539\times 10^{58}$
Root discriminant \(123.32\)
Ramified primes $23,29$
Class number $1$ (GRH)
Class group trivial (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 - 173*x^26 + 173*x^25 + 13399*x^24 - 13399*x^23 - 613001*x^22 + 613001*x^21 + 18404503*x^20 - 18404503*x^19 - 380963081*x^18 + 380963081*x^17 + 5557459255*x^16 - 5557459255*x^15 - 57374393033*x^14 + 57374393033*x^13 + 414614499127*x^12 - 414614499127*x^11 - 2039727740105*x^10 + 2039727740105*x^9 + 6485882143543*x^8 - 6485882143543*x^7 - 12115448511689*x^6 + 12115448511689*x^5 + 10862665827127*x^4 - 10862665827127*x^3 - 2393938599113*x^2 + 2393938599113*x - 121377840329)
 
gp: K = bnfinit(y^28 - y^27 - 173*y^26 + 173*y^25 + 13399*y^24 - 13399*y^23 - 613001*y^22 + 613001*y^21 + 18404503*y^20 - 18404503*y^19 - 380963081*y^18 + 380963081*y^17 + 5557459255*y^16 - 5557459255*y^15 - 57374393033*y^14 + 57374393033*y^13 + 414614499127*y^12 - 414614499127*y^11 - 2039727740105*y^10 + 2039727740105*y^9 + 6485882143543*y^8 - 6485882143543*y^7 - 12115448511689*y^6 + 12115448511689*y^5 + 10862665827127*y^4 - 10862665827127*y^3 - 2393938599113*y^2 + 2393938599113*y - 121377840329, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - x^27 - 173*x^26 + 173*x^25 + 13399*x^24 - 13399*x^23 - 613001*x^22 + 613001*x^21 + 18404503*x^20 - 18404503*x^19 - 380963081*x^18 + 380963081*x^17 + 5557459255*x^16 - 5557459255*x^15 - 57374393033*x^14 + 57374393033*x^13 + 414614499127*x^12 - 414614499127*x^11 - 2039727740105*x^10 + 2039727740105*x^9 + 6485882143543*x^8 - 6485882143543*x^7 - 12115448511689*x^6 + 12115448511689*x^5 + 10862665827127*x^4 - 10862665827127*x^3 - 2393938599113*x^2 + 2393938599113*x - 121377840329);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - x^27 - 173*x^26 + 173*x^25 + 13399*x^24 - 13399*x^23 - 613001*x^22 + 613001*x^21 + 18404503*x^20 - 18404503*x^19 - 380963081*x^18 + 380963081*x^17 + 5557459255*x^16 - 5557459255*x^15 - 57374393033*x^14 + 57374393033*x^13 + 414614499127*x^12 - 414614499127*x^11 - 2039727740105*x^10 + 2039727740105*x^9 + 6485882143543*x^8 - 6485882143543*x^7 - 12115448511689*x^6 + 12115448511689*x^5 + 10862665827127*x^4 - 10862665827127*x^3 - 2393938599113*x^2 + 2393938599113*x - 121377840329)
 

\( x^{28} - x^{27} - 173 x^{26} + 173 x^{25} + 13399 x^{24} - 13399 x^{23} - 613001 x^{22} + \cdots - 121377840329 \) 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:  $[28, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(35394489068231220324814698212289719250778220848093751207381\) \(\medspace = 23^{14}\cdot 29^{27}\) 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:  \(123.32\)
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:  $23^{1/2}29^{27/28}\approx 123.31997158416979$
Ramified primes:   \(23\), \(29\) 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:  \(667=23\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{667}(576,·)$, $\chi_{667}(1,·)$, $\chi_{667}(323,·)$, $\chi_{667}(68,·)$, $\chi_{667}(645,·)$, $\chi_{667}(321,·)$, $\chi_{667}(137,·)$, $\chi_{667}(139,·)$, $\chi_{667}(206,·)$, $\chi_{667}(208,·)$, $\chi_{667}(275,·)$, $\chi_{667}(277,·)$, $\chi_{667}(24,·)$, $\chi_{667}(93,·)$, $\chi_{667}(415,·)$, $\chi_{667}(160,·)$, $\chi_{667}(482,·)$, $\chi_{667}(484,·)$, $\chi_{667}(229,·)$, $\chi_{667}(231,·)$, $\chi_{667}(298,·)$, $\chi_{667}(620,·)$, $\chi_{667}(622,·)$, $\chi_{667}(367,·)$, $\chi_{667}(114,·)$, $\chi_{667}(505,·)$, $\chi_{667}(254,·)$, $\chi_{667}(597,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{75943352179}a^{15}+\frac{23960563227}{75943352179}a^{14}-\frac{90}{75943352179}a^{13}+\frac{37783197765}{75943352179}a^{12}+\frac{3240}{75943352179}a^{11}-\frac{31751891381}{75943352179}a^{10}-\frac{59400}{75943352179}a^{9}-\frac{25834943051}{75943352179}a^{8}+\frac{583200}{75943352179}a^{7}+\frac{19560805963}{75943352179}a^{6}-\frac{2939328}{75943352179}a^{5}-\frac{2299871673}{75943352179}a^{4}+\frac{6531840}{75943352179}a^{3}-\frac{34521868580}{75943352179}a^{2}-\frac{4199040}{75943352179}a+\frac{13103928153}{75943352179}$, $\frac{1}{75943352179}a^{16}-\frac{96}{75943352179}a^{14}-\frac{8123324996}{75943352179}a^{13}+\frac{3744}{75943352179}a^{12}+\frac{26072532256}{75943352179}a^{11}-\frac{76032}{75943352179}a^{10}-\frac{22742445890}{75943352179}a^{9}+\frac{855360}{75943352179}a^{8}+\frac{23717812100}{75943352179}a^{7}-\frac{5225472}{75943352179}a^{6}-\frac{14137980342}{75943352179}a^{5}+\frac{15676416}{75943352179}a^{4}-\frac{33529411153}{75943352179}a^{3}-\frac{17915904}{75943352179}a^{2}+\frac{28739495274}{75943352179}a+\frac{3359232}{75943352179}$, $\frac{1}{75943352179}a^{17}+\frac{13790179426}{75943352179}a^{14}-\frac{4896}{75943352179}a^{13}+\frac{7978613104}{75943352179}a^{12}+\frac{235008}{75943352179}a^{11}-\frac{33189931306}{75943352179}a^{10}-\frac{4847040}{75943352179}a^{9}-\frac{26249451068}{75943352179}a^{8}+\frac{50761728}{75943352179}a^{7}-\frac{34884412369}{75943352179}a^{6}-\frac{266499072}{75943352179}a^{5}-\frac{26487035224}{75943352179}a^{4}+\frac{609140736}{75943352179}a^{3}-\frac{19795744709}{75943352179}a^{2}-\frac{399748608}{75943352179}a-\frac{33059884355}{75943352179}$, $\frac{1}{75943352179}a^{18}-\frac{5508}{75943352179}a^{14}+\frac{34001126580}{75943352179}a^{13}+\frac{286416}{75943352179}a^{12}+\frac{17263161885}{75943352179}a^{11}-\frac{6543504}{75943352179}a^{10}-\frac{14588149362}{75943352179}a^{9}+\frac{78522048}{75943352179}a^{8}+\frac{9413452710}{75943352179}a^{7}-\frac{499685760}{75943352179}a^{6}-\frac{18880484598}{75943352179}a^{5}+\frac{1541887488}{75943352179}a^{4}+\frac{5491560666}{75943352179}a^{3}-\frac{1798868736}{75943352179}a^{2}-\frac{33042433772}{75943352179}a+\frac{342641664}{75943352179}$, $\frac{1}{75943352179}a^{19}+\frac{19237293794}{75943352179}a^{14}-\frac{209304}{75943352179}a^{13}-\frac{33611871134}{75943352179}a^{12}+\frac{11302416}{75943352179}a^{11}-\frac{6465807673}{75943352179}a^{10}-\frac{248653152}{75943352179}a^{9}+\frac{28389111248}{75943352179}a^{8}+\frac{2712579840}{75943352179}a^{7}+\frac{34365369784}{75943352179}a^{6}-\frac{14647931136}{75943352179}a^{5}+\frac{20338199675}{75943352179}a^{4}+\frac{34178505984}{75943352179}a^{3}-\frac{17340716196}{75943352179}a^{2}-\frac{22785670656}{75943352179}a+\frac{30251696674}{75943352179}$, $\frac{1}{75943352179}a^{20}-\frac{246240}{75943352179}a^{14}+\frac{26990822388}{75943352179}a^{13}+\frac{14405040}{75943352179}a^{12}+\frac{14194438726}{75943352179}a^{11}-\frac{351039744}{75943352179}a^{10}+\frac{4020237435}{75943352179}a^{9}+\frac{4387996800}{75943352179}a^{8}+\frac{31985464833}{75943352179}a^{7}-\frac{28721433600}{75943352179}a^{6}-\frac{25384027028}{75943352179}a^{5}+\frac{14529163661}{75943352179}a^{4}+\frac{16759030096}{75943352179}a^{3}-\frac{31283333261}{75943352179}a^{2}-\frac{13367669422}{75943352179}a+\frac{20679432192}{75943352179}$, $\frac{1}{75943352179}a^{21}+\frac{37049052358}{75943352179}a^{14}-\frac{7756560}{75943352179}a^{13}+\frac{4679995215}{75943352179}a^{12}+\frac{446777856}{75943352179}a^{11}+\frac{14223464582}{75943352179}a^{10}-\frac{10238659200}{75943352179}a^{9}-\frac{17609435114}{75943352179}a^{8}-\frac{37000969958}{75943352179}a^{7}-\frac{3692298804}{75943352179}a^{6}-\frac{25760793448}{75943352179}a^{5}+\frac{5935469379}{75943352179}a^{4}-\frac{17693447420}{75943352179}a^{3}-\frac{35104004436}{75943352179}a^{2}-\frac{26028599081}{75943352179}a+\frac{30121013368}{75943352179}$, $\frac{1}{75943352179}a^{22}-\frac{9480240}{75943352179}a^{14}-\frac{2412788441}{75943352179}a^{13}+\frac{591566976}{75943352179}a^{12}-\frac{34209732518}{75943352179}a^{11}-\frac{15016700160}{75943352179}a^{10}+\frac{9641187024}{75943352179}a^{9}-\frac{34758197337}{75943352179}a^{8}+\frac{11817723781}{75943352179}a^{7}+\frac{965927843}{75943352179}a^{6}-\frac{26669033142}{75943352179}a^{5}+\frac{27286371774}{75943352179}a^{4}-\frac{13278455842}{75943352179}a^{3}-\frac{17554975693}{75943352179}a^{2}-\frac{5607797065}{75943352179}a-\frac{22223409367}{75943352179}$, $\frac{1}{75943352179}a^{23}+\frac{5112304509}{75943352179}a^{14}-\frac{261654624}{75943352179}a^{13}+\frac{17172629293}{75943352179}a^{12}+\frac{15699277440}{75943352179}a^{11}-\frac{15223004801}{75943352179}a^{10}+\frac{9662364095}{75943352179}a^{9}-\frac{36973795793}{75943352179}a^{8}-\frac{14022813224}{75943352179}a^{7}-\frac{26914370487}{75943352179}a^{6}+\frac{32961742747}{75943352179}a^{5}-\frac{12297106462}{75943352179}a^{4}+\frac{12023840022}{75943352179}a^{3}-\frac{7079821135}{75943352179}a^{2}-\frac{35813837171}{75943352179}a+\frac{20508731983}{75943352179}$, $\frac{1}{75943352179}a^{24}-\frac{330511104}{75943352179}a^{14}+\frac{21619922029}{75943352179}a^{13}+\frac{21483221760}{75943352179}a^{12}-\frac{23438838939}{75943352179}a^{11}-\frac{29321094107}{75943352179}a^{10}+\frac{12392027165}{75943352179}a^{9}-\frac{4370319763}{75943352179}a^{8}+\frac{13102528253}{75943352179}a^{7}-\frac{2553191018}{75943352179}a^{6}-\frac{31718230882}{75943352179}a^{5}+\frac{2196870724}{75943352179}a^{4}-\frac{14550668321}{75943352179}a^{3}-\frac{21073383224}{75943352179}a^{2}+\frac{36160469771}{75943352179}a-\frac{28407736372}{75943352179}$, $\frac{1}{75943352179}a^{25}+\frac{32156257966}{75943352179}a^{14}-\frac{8262777600}{75943352179}a^{13}+\frac{2829168501}{75943352179}a^{12}-\frac{21672047653}{75943352179}a^{11}-\frac{19075344386}{75943352179}a^{10}+\frac{32598316998}{75943352179}a^{9}-\frac{3580311620}{75943352179}a^{8}+\frac{7294831480}{75943352179}a^{7}-\frac{28280409871}{75943352179}a^{6}-\frac{10984353620}{75943352179}a^{5}-\frac{28301952365}{75943352179}a^{4}-\frac{17096224297}{75943352179}a^{3}-\frac{19155956342}{75943352179}a^{2}+\frac{7007194693}{75943352179}a-\frac{23649864060}{75943352179}$, $\frac{1}{75943352179}a^{26}-\frac{10741610880}{75943352179}a^{14}+\frac{11045002639}{75943352179}a^{13}+\frac{34663243509}{75943352179}a^{12}-\frac{11071964638}{75943352179}a^{11}-\frac{3825839052}{75943352179}a^{10}+\frac{26892214751}{75943352179}a^{9}-\frac{24932164819}{75943352179}a^{8}-\frac{30595746632}{75943352179}a^{7}+\frac{1515017883}{75943352179}a^{6}+\frac{31971090305}{75943352179}a^{5}+\frac{8849641870}{75943352179}a^{4}+\frac{15666953678}{75943352179}a^{3}+\frac{6060071532}{75943352179}a^{2}+\frac{8209231055}{75943352179}a-\frac{35414931065}{75943352179}$, $\frac{1}{75943352179}a^{27}+\frac{11730835786}{75943352179}a^{14}-\frac{20761509543}{75943352179}a^{13}+\frac{7261403193}{75943352179}a^{12}+\frac{16938114166}{75943352179}a^{11}+\frac{29437893760}{75943352179}a^{10}-\frac{573428861}{75943352179}a^{9}-\frac{3212901162}{75943352179}a^{8}+\frac{17802340352}{75943352179}a^{7}+\frac{29785244894}{75943352179}a^{6}+\frac{36119963764}{75943352179}a^{5}+\frac{24410335322}{75943352179}a^{4}+\frac{21402708391}{75943352179}a^{3}-\frac{4905841411}{75943352179}a^{2}+\frac{14391721952}{75943352179}a-\frac{9187060160}{75943352179}$ 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

Trivial group, which has order $1$ (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:  $27$
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:   $\frac{85}{75943352179}a^{23}-\frac{11730}{75943352179}a^{21}+\frac{703800}{75943352179}a^{19}-\frac{24069960}{75943352179}a^{17}+\frac{516870720}{75943352179}a^{15}-\frac{7236190080}{75943352179}a^{13}+\frac{66402685440}{75943352179}a^{11}-\frac{391301539200}{75943352179}a^{9}+\frac{1408685541120}{75943352179}a^{7}+\frac{5816861}{75943352179}a^{6}-\frac{2817371082240}{75943352179}a^{5}-\frac{209406996}{75943352179}a^{4}+\frac{2600650229760}{75943352179}a^{3}+\frac{1884662964}{75943352179}a^{2}-\frac{709268244480}{75943352179}a-\frac{2512883952}{75943352179}$, $\frac{1}{75943352179}a^{27}-\frac{162}{75943352179}a^{25}+\frac{11664}{75943352179}a^{23}-\frac{491832}{75943352179}a^{21}+\frac{13471920}{75943352179}a^{19}-\frac{251312544}{75943352179}a^{17}+\frac{3255095808}{75943352179}a^{15}-\frac{29295862272}{75943352179}a^{13}+\frac{180400836096}{75943352179}a^{11}-\frac{734966369280}{75943352179}a^{9}+\frac{1867679244288}{75943352179}a^{7}-\frac{2674177099776}{75943352179}a^{5}+\frac{1782784733184}{75943352179}a^{3}+\frac{9916653245}{75943352179}a^{2}-\frac{352638738432}{75943352179}a-\frac{118999838940}{75943352179}$, $\frac{11}{75943352179}a^{25}-\frac{1650}{75943352179}a^{23}+\frac{108900}{75943352179}a^{21}-\frac{4158000}{75943352179}a^{19}+\frac{101574000}{75943352179}a^{17}-\frac{1657687680}{75943352179}a^{15}+\frac{18321811200}{75943352179}a^{13}-\frac{136104883200}{75943352179}a^{11}+\frac{660508992000}{75943352179}a^{9}-\frac{1981526976000}{75943352179}a^{7}+\frac{3328965319680}{75943352179}a^{5}+\frac{920550931}{75943352179}a^{4}-\frac{2593998950400}{75943352179}a^{3}-\frac{22093222344}{75943352179}a^{2}+\frac{598615142400}{75943352179}a+\frac{142223019211}{75943352179}$, $\frac{11}{75943352179}a^{25}-\frac{1650}{75943352179}a^{23}+\frac{108900}{75943352179}a^{21}-\frac{4158000}{75943352179}a^{19}+\frac{101574000}{75943352179}a^{17}-\frac{1657687680}{75943352179}a^{15}+\frac{18321811200}{75943352179}a^{13}-\frac{136104883200}{75943352179}a^{11}+\frac{660508992000}{75943352179}a^{9}-\frac{1981526976000}{75943352179}a^{7}+\frac{3328965319680}{75943352179}a^{5}+\frac{920550931}{75943352179}a^{4}-\frac{2593998950400}{75943352179}a^{3}-\frac{22093222344}{75943352179}a^{2}+\frac{598615142400}{75943352179}a+\frac{66279667032}{75943352179}$, $\frac{12155}{75943352179}a^{17}-\frac{1239810}{75943352179}a^{15}+\frac{52072020}{75943352179}a^{13}-\frac{1609901}{75943352179}a^{12}-\frac{1160462160}{75943352179}a^{11}+\frac{115912872}{75943352179}a^{10}+\frac{14728942800}{75943352179}a^{9}-\frac{3129647544}{75943352179}a^{8}-\frac{106048388160}{75943352179}a^{7}+\frac{38946724992}{75943352179}a^{6}+\frac{404912027520}{75943352179}a^{5}-\frac{219075328080}{75943352179}a^{4}-\frac{694134904320}{75943352179}a^{3}+\frac{450669246336}{75943352179}a^{2}+\frac{347067452160}{75943352179}a-\frac{150223082112}{75943352179}$, $\frac{1}{75943352179}a^{27}+\frac{5}{75943352179}a^{26}-\frac{162}{75943352179}a^{25}-\frac{780}{75943352179}a^{24}+\frac{11749}{75943352179}a^{23}+\frac{53849}{75943352179}a^{22}-\frac{503562}{75943352179}a^{21}-\frac{2165988}{75943352179}a^{20}+\frac{14178589}{75943352179}a^{19}+\frac{56232737}{75943352179}a^{18}-\frac{275709570}{75943352179}a^{17}-\frac{986024556}{75943352179}a^{16}+\frac{3787696117}{75943352179}a^{15}+\frac{11870204585}{75943352179}a^{14}-\frac{36946893402}{75943352179}a^{13}-\frac{97695953268}{75943352179}a^{12}+\frac{253328667277}{75943352179}a^{11}+\frac{536470254065}{75943352179}a^{10}-\frac{1188542382258}{75943352179}a^{9}-\frac{1866565961724}{75943352179}a^{8}+\frac{3626140232869}{75943352179}a^{7}+\frac{3718816451513}{75943352179}a^{6}-\frac{6544808878218}{75943352179}a^{5}-\frac{3365624106756}{75943352179}a^{4}+\frac{5717443768381}{75943352179}a^{3}+\frac{481659202049}{75943352179}a^{2}-\frac{1244721851874}{75943352179}a+\frac{104213219700}{75943352179}$, $\frac{42509}{75943352179}a^{16}+\frac{30421}{75943352179}a^{15}-\frac{4366339}{75943352179}a^{14}-\frac{2634941}{75943352179}a^{13}+\frac{183133596}{75943352179}a^{12}+\frac{90534018}{75943352179}a^{11}-\frac{4023380988}{75943352179}a^{10}-\frac{1566106740}{75943352179}a^{9}+\frac{49309644240}{75943352179}a^{8}+\frac{14272557696}{75943352179}a^{7}-\frac{330902415648}{75943352179}a^{6}-\frac{65134510560}{75943352179}a^{5}+\frac{1101480073344}{75943352179}a^{4}+\frac{125856745056}{75943352179}a^{3}-\frac{1414224121536}{75943352179}a^{2}-\frac{65297544768}{75943352179}a+\frac{302627052288}{75943352179}$, $\frac{19}{75943352179}a^{24}-\frac{2736}{75943352179}a^{22}+\frac{172368}{75943352179}a^{20}-\frac{6238080}{75943352179}a^{18}+\frac{143163936}{75943352179}a^{16}-\frac{2170063872}{75943352179}a^{14}+\frac{21941756928}{75943352179}a^{12}-\frac{146032533504}{75943352179}a^{10}+\frac{616074750720}{75943352179}a^{8}-\frac{1533341601792}{75943352179}a^{6}-\frac{885649765}{75943352179}a^{5}+\frac{1971439202304}{75943352179}a^{4}+\frac{26569492950}{75943352179}a^{3}-\frac{992612745216}{75943352179}a^{2}-\frac{159416957700}{75943352179}a+\frac{158661080947}{75943352179}$, $\frac{12155}{75943352179}a^{17}-\frac{1239810}{75943352179}a^{15}+\frac{52072020}{75943352179}a^{13}-\frac{1609901}{75943352179}a^{12}-\frac{1160462160}{75943352179}a^{11}+\frac{115912872}{75943352179}a^{10}+\frac{14728942800}{75943352179}a^{9}-\frac{3129647544}{75943352179}a^{8}-\frac{106048388160}{75943352179}a^{7}+\frac{38946724992}{75943352179}a^{6}+\frac{404912027520}{75943352179}a^{5}-\frac{219075328080}{75943352179}a^{4}-\frac{694134904320}{75943352179}a^{3}+\frac{450669246336}{75943352179}a^{2}+\frac{347067452160}{75943352179}a-\frac{74279729933}{75943352179}$, $\frac{5}{75943352179}a^{26}-\frac{780}{75943352179}a^{24}+\frac{53820}{75943352179}a^{22}-\frac{2162160}{75943352179}a^{20}+\frac{56019600}{75943352179}a^{18}-\frac{979542720}{75943352179}a^{16}+\frac{11754512640}{75943352179}a^{14}-\frac{96510735360}{75943352179}a^{12}+\frac{530809044480}{75943352179}a^{10}-\frac{1873443686400}{75943352179}a^{8}+\frac{3934231741440}{75943352179}a^{6}-\frac{4291889172480}{75943352179}a^{4}-\frac{4393347659}{75943352179}a^{3}+\frac{1839381073920}{75943352179}a^{2}+\frac{79080257862}{75943352179}a-\frac{130606940160}{75943352179}$, $\frac{539}{75943352179}a^{21}-\frac{67914}{75943352179}a^{19}+\frac{3667356}{75943352179}a^{17}-\frac{110835648}{75943352179}a^{15}+\frac{2053719360}{75943352179}a^{13}-\frac{24028516512}{75943352179}a^{11}+\frac{176209121088}{75943352179}a^{9}-\frac{23793485}{75943352179}a^{8}-\frac{776758574592}{75943352179}a^{7}+\frac{1142087280}{75943352179}a^{6}+\frac{1882145776896}{75943352179}a^{5}-\frac{17131309200}{75943352179}a^{4}-\frac{2091273085440}{75943352179}a^{3}+\frac{82230284160}{75943352179}a^{2}+\frac{684416646144}{75943352179}a-\frac{61672713120}{75943352179}$, $\frac{5059}{75943352179}a^{18}-\frac{546372}{75943352179}a^{16}+\frac{24586740}{75943352179}a^{14}-\frac{596638224}{75943352179}a^{12}+\frac{2227595}{75943352179}a^{11}+\frac{8438169168}{75943352179}a^{10}-\frac{147021270}{75943352179}a^{9}-\frac{70101713088}{75943352179}a^{8}+\frac{3528510480}{75943352179}a^{7}+\frac{327141327744}{75943352179}a^{6}-\frac{37049360040}{75943352179}a^{5}-\frac{764745960960}{75943352179}a^{4}+\frac{158782971600}{75943352179}a^{3}+\frac{688271364864}{75943352179}a^{2}-\frac{190539565920}{75943352179}a-\frac{26022775949}{75943352179}$, $\frac{29}{75943352179}a^{22}-\frac{3828}{75943352179}a^{20}+\frac{218196}{75943352179}a^{18}-\frac{12155}{75943352179}a^{17}-\frac{7028208}{75943352179}a^{16}+\frac{1239810}{75943352179}a^{15}+\frac{140564160}{75943352179}a^{14}-\frac{52072020}{75943352179}a^{13}-\frac{1804226131}{75943352179}a^{12}+\frac{1160462160}{75943352179}a^{11}+\frac{14782234392}{75943352179}a^{10}-\frac{14728942800}{75943352179}a^{9}-\frac{73489395528}{75943352179}a^{8}+\frac{105899810389}{75943352179}a^{7}+\frac{190910404224}{75943352179}a^{6}-\frac{398671761138}{75943352179}a^{5}-\frac{134551024560}{75943352179}a^{4}+\frac{619251707736}{75943352179}a^{3}-\frac{238493434752}{75943352179}a^{2}-\frac{122417862408}{75943352179}a+\frac{129180852864}{75943352179}$, $\frac{2}{75943352179}a^{27}+\frac{5}{75943352179}a^{26}-\frac{283}{75943352179}a^{25}-\frac{799}{75943352179}a^{24}+\frac{17263}{75943352179}a^{23}+\frac{56701}{75943352179}a^{22}-\frac{590033}{75943352179}a^{21}-\frac{2353303}{75943352179}a^{20}+\frac{12206757}{75943352179}a^{19}+\frac{63299801}{75943352179}a^{18}-\frac{150549701}{75943352179}a^{17}-\frac{1155043348}{75943352179}a^{16}+\frac{901432951}{75943352179}a^{15}+\frac{14537132201}{75943352179}a^{14}+\frac{1908455984}{75943352179}a^{13}-\frac{125692763479}{75943352179}a^{12}-\frac{79287355943}{75943352179}a^{11}+\frac{728785931024}{75943352179}a^{10}+\frac{644696012347}{75943352179}a^{9}-\frac{2692763754523}{75943352179}a^{8}-\frac{2732255759125}{75943352179}a^{7}+\frac{5746672044605}{75943352179}a^{6}+\frac{6601881058255}{75943352179}a^{5}-\frac{5730509998252}{75943352179}a^{4}-\frac{8955336402569}{75943352179}a^{3}+\frac{1371696199918}{75943352179}a^{2}+\frac{5798276475589}{75943352179}a-\frac{309286314427}{75943352179}$, $\frac{6}{75943352179}a^{26}-\frac{936}{75943352179}a^{24}+\frac{255}{75943352179}a^{23}+\frac{64045}{75943352179}a^{22}-\frac{35190}{75943352179}a^{21}-\frac{2523809}{75943352179}a^{20}+\frac{2110721}{75943352179}a^{19}+\frac{63211884}{75943352179}a^{18}-\frac{72132474}{75943352179}a^{17}-\frac{1047045248}{75943352179}a^{16}+\frac{1546896672}{75943352179}a^{15}+\frac{11554772160}{75943352179}a^{14}-\frac{21609222881}{75943352179}a^{13}-\frac{83280148848}{75943352179}a^{12}+\frac{197544931662}{75943352179}a^{11}+\frac{370544273288}{75943352179}a^{10}-\frac{1155330946153}{75943352179}a^{9}-\frac{888485494368}{75943352179}a^{8}+\frac{4086685284348}{75943352179}a^{7}+\frac{691072986039}{75943352179}a^{6}-\frac{7811672131368}{75943352179}a^{5}+\frac{800524201284}{75943352179}a^{4}+\frac{6408666554621}{75943352179}a^{3}-\frac{664737977700}{75943352179}a^{2}-\frac{1533394347786}{75943352179}a+\frac{88047348988}{75943352179}$, $\frac{1}{75943352179}a^{27}+\frac{10}{75943352179}a^{26}-\frac{184}{75943352179}a^{25}-\frac{1598}{75943352179}a^{24}+\frac{15134}{75943352179}a^{23}+\frac{113170}{75943352179}a^{22}-\frac{733805}{75943352179}a^{21}-\frac{4673113}{75943352179}a^{20}+\frac{23291775}{75943352179}a^{19}+\frac{124494577}{75943352179}a^{18}-\frac{508207564}{75943352179}a^{17}-\frac{2234626970}{75943352179}a^{16}+\frac{7788482510}{75943352179}a^{15}+\frac{27376707766}{75943352179}a^{14}-\frac{84159717754}{75943352179}a^{13}-\frac{226552686926}{75943352179}a^{12}+\frac{634071341789}{75943352179}a^{11}+\frac{1220911684826}{75943352179}a^{10}-\frac{3239112851074}{75943352179}a^{9}-\frac{3953774761520}{75943352179}a^{8}+\frac{10649354070794}{75943352179}a^{7}+\frac{6297124179730}{75943352179}a^{6}-\frac{20452718744050}{75943352179}a^{5}-\frac{1194745230734}{75943352179}a^{4}+\frac{18664009477790}{75943352179}a^{3}-\frac{6643816899499}{75943352179}a^{2}-\frac{4007210253353}{75943352179}a+\frac{2394300150266}{75943352179}$, $\frac{365}{75943352179}a^{20}-\frac{5059}{75943352179}a^{19}-\frac{16468}{75943352179}a^{18}+\frac{528106}{75943352179}a^{17}-\frac{815162}{75943352179}a^{16}-\frac{22122237}{75943352179}a^{15}+\frac{78433169}{75943352179}a^{14}+\frac{462241593}{75943352179}a^{13}-\frac{2449741744}{75943352179}a^{12}-\frac{4591119831}{75943352179}a^{11}+\frac{39327036594}{75943352179}a^{10}+\frac{7732750915}{75943352179}a^{9}-\frac{345910793688}{75943352179}a^{8}+\frac{245228174082}{75943352179}a^{7}+\frac{1584614120328}{75943352179}a^{6}-\frac{2018504163852}{75943352179}a^{5}-\frac{3136038423264}{75943352179}a^{4}+\frac{5366547797472}{75943352179}a^{3}+\frac{1367474444064}{75943352179}a^{2}-\frac{3673799407440}{75943352179}a+\frac{340017077376}{75943352179}$, $\frac{4}{75943352179}a^{27}-\frac{637}{75943352179}a^{25}+\frac{45006}{75943352179}a^{23}+\frac{87}{75943352179}a^{22}-\frac{1858428}{75943352179}a^{21}-\frac{13988}{75943352179}a^{20}+\frac{49729680}{75943352179}a^{19}+\frac{972282}{75943352179}a^{18}-\frac{903676176}{75943352179}a^{17}-\frac{38341146}{75943352179}a^{16}+\frac{11362695552}{75943352179}a^{15}+\frac{945045000}{75943352179}a^{14}-\frac{98860027987}{75943352179}a^{13}-\frac{15103501920}{75943352179}a^{12}+\frac{585365457095}{75943352179}a^{11}+\frac{156913881696}{75943352179}a^{10}-\frac{2275033429007}{75943352179}a^{9}-\frac{1031956257024}{75943352179}a^{8}+\frac{5419194089781}{75943352179}a^{7}+\frac{4034345812992}{75943352179}a^{6}-\frac{6782128454970}{75943352179}a^{5}-\frac{8275152142829}{75943352179}a^{4}+\frac{2219843357928}{75943352179}a^{3}+\frac{6750165126073}{75943352179}a^{2}+\frac{2236834890360}{75943352179}a-\frac{152511058664}{75943352179}$, $\frac{85}{75943352179}a^{24}+\frac{340}{75943352179}a^{23}-\frac{11133}{75943352179}a^{22}-\frac{45129}{75943352179}a^{21}+\frac{627135}{75943352179}a^{20}+\frac{2592403}{75943352179}a^{19}-\frac{19834812}{75943352179}a^{18}-\frac{84420942}{75943352179}a^{17}+\frac{385277256}{75943352179}a^{16}+\frac{1714895136}{75943352179}a^{15}-\frac{4712126400}{75943352179}a^{14}-\frac{22532723640}{75943352179}a^{13}+\frac{35533998864}{75943352179}a^{12}+\frac{192197114928}{75943352179}a^{11}-\frac{151210616515}{75943352179}a^{10}-\frac{1040395631804}{75943352179}a^{9}+\frac{259745938018}{75943352179}a^{8}+\frac{3393644141665}{75943352179}a^{7}+\frac{334945122320}{75943352179}a^{6}-\frac{6088091000700}{75943352179}a^{5}-\frac{1794142656864}{75943352179}a^{4}+\frac{5165689664916}{75943352179}a^{3}+\frac{2029931341344}{75943352179}a^{2}-\frac{1454853292560}{75943352179}a-\frac{639109116864}{75943352179}$, $\frac{1}{75943352179}a^{27}+\frac{5}{75943352179}a^{26}-\frac{143}{75943352179}a^{25}-\frac{903}{75943352179}a^{24}+\frac{8899}{75943352179}a^{23}+\frac{71561}{75943352179}a^{22}-\frac{316001}{75943352179}a^{21}-\frac{3280384}{75943352179}a^{20}+\frac{7059444}{75943352179}a^{19}+\frac{96440897}{75943352179}a^{18}-\frac{103366355}{75943352179}a^{17}-\frac{1903844959}{75943352179}a^{16}+\frac{1008494111}{75943352179}a^{15}+\frac{25662611946}{75943352179}a^{14}-\frac{6650711633}{75943352179}a^{13}-\frac{235313843131}{75943352179}a^{12}+\frac{31059733831}{75943352179}a^{11}+\frac{1434317484635}{75943352179}a^{10}-\frac{112629489434}{75943352179}a^{9}-\frac{5543526809455}{75943352179}a^{8}+\frac{302147918989}{75943352179}a^{7}+\frac{12465461043281}{75943352179}a^{6}-\frac{270524250910}{75943352179}a^{5}-\frac{13776650260386}{75943352179}a^{4}-\frac{1126777981163}{75943352179}a^{3}+\frac{4906097124593}{75943352179}a^{2}+\frac{2398219581006}{75943352179}a-\frac{108950565891}{75943352179}$, $\frac{3}{75943352179}a^{27}+\frac{15}{75943352179}a^{26}-\frac{492}{75943352179}a^{25}-\frac{2397}{75943352179}a^{24}+\frac{36147}{75943352179}a^{23}+\frac{169216}{75943352179}a^{22}-\frac{1571703}{75943352179}a^{21}-\frac{6941365}{75943352179}a^{20}+\frac{45007509}{75943352179}a^{19}+\frac{183050415}{75943352179}a^{18}-\frac{893492269}{75943352179}a^{17}-\frac{3241173621}{75943352179}a^{16}+\frac{12595483078}{75943352179}a^{15}+\frac{39054218744}{75943352179}a^{14}-\frac{126815873973}{75943352179}a^{13}-\frac{317314267794}{75943352179}a^{12}+\frac{902326633293}{75943352179}a^{11}+\frac{1680526181031}{75943352179}a^{10}-\frac{4403960159951}{75943352179}a^{9}-\frac{5389167849069}{75943352179}a^{8}+\frac{13918268205915}{75943352179}a^{7}+\frac{8802546583227}{75943352179}a^{6}-\frac{25558989665271}{75943352179}a^{5}-\frac{3342799461892}{75943352179}a^{4}+\frac{21442533840885}{75943352179}a^{3}-\frac{5109143084925}{75943352179}a^{2}-\frac{2804163652583}{75943352179}a+\frac{337292377251}{75943352179}$, $\frac{11}{75943352179}a^{27}+\frac{5}{75943352179}a^{26}-\frac{1892}{75943352179}a^{25}-\frac{875}{75943352179}a^{24}+\frac{145314}{75943352179}a^{23}+\frac{67587}{75943352179}a^{22}-\frac{6572227}{75943352179}a^{21}-\frac{3027191}{75943352179}a^{20}+\frac{194352894}{75943352179}a^{19}+\frac{86839074}{75943352179}a^{18}-\frac{3945304528}{75943352179}a^{17}-\frac{1662057227}{75943352179}a^{16}+\frac{56160968949}{75943352179}a^{15}+\frac{21410813284}{75943352179}a^{14}-\frac{562656047651}{75943352179}a^{13}-\frac{182192533230}{75943352179}a^{12}+\frac{3923896204404}{75943352179}a^{11}+\frac{966864514099}{75943352179}a^{10}-\frac{18545888341193}{75943352179}a^{9}-\frac{2744519338691}{75943352179}a^{8}+\frac{56612676406485}{75943352179}a^{7}+\frac{1799501855490}{75943352179}a^{6}-\frac{102449476807513}{75943352179}a^{5}+\frac{9115362537770}{75943352179}a^{4}+\frac{92415496628767}{75943352179}a^{3}-\frac{16466659659804}{75943352179}a^{2}-\frac{25435987032923}{75943352179}a+\frac{2262658466114}{75943352179}$, $\frac{6}{75943352179}a^{27}-\frac{27}{75943352179}a^{26}-\frac{961}{75943352179}a^{25}+\frac{4212}{75943352179}a^{24}+\frac{68334}{75943352179}a^{23}-\frac{290628}{75943352179}a^{22}-\frac{2843535}{75943352179}a^{21}+\frac{11673890}{75943352179}a^{20}+\frac{76855338}{75943352179}a^{19}-\frac{302292960}{75943352179}a^{18}-\frac{1416119436}{75943352179}a^{17}+\frac{5278673808}{75943352179}a^{16}+\frac{18169492460}{75943352179}a^{15}-\frac{63166679156}{75943352179}a^{14}-\frac{162940587192}{75943352179}a^{13}+\frac{515831589744}{75943352179}a^{12}+\frac{1010234540160}{75943352179}a^{11}-\frac{2807969818896}{75943352179}a^{10}-\frac{4213697739327}{75943352179}a^{9}+\frac{9709648503484}{75943352179}a^{8}+\frac{11227327790346}{75943352179}a^{7}-\frac{19495013562048}{75943352179}a^{6}-\frac{17293283567028}{75943352179}a^{5}+\frac{18922972795603}{75943352179}a^{4}+\frac{12206700269591}{75943352179}a^{3}-\frac{5306069426125}{75943352179}a^{2}-\frac{1470959275950}{75943352179}a-\frac{85068935916}{75943352179}$, $\frac{3}{75943352179}a^{27}-\frac{20}{75943352179}a^{26}-\frac{549}{75943352179}a^{25}+\frac{3025}{75943352179}a^{24}+\frac{44527}{75943352179}a^{23}-\frac{202110}{75943352179}a^{22}-\frac{2113795}{75943352179}a^{21}+\frac{7847652}{75943352179}a^{20}+\frac{65295054}{75943352179}a^{19}-\frac{195954139}{75943352179}a^{18}-\frac{1379343131}{75943352179}a^{17}+\frac{3286882511}{75943352179}a^{16}+\frac{20373037289}{75943352179}a^{15}-\frac{37543637947}{75943352179}a^{14}-\frac{211251149688}{75943352179}a^{13}+\frac{289494923923}{75943352179}a^{12}+\frac{1520083680961}{75943352179}a^{11}-\frac{1459393309332}{75943352179}a^{10}-\frac{7369027592528}{75943352179}a^{9}+\frac{4501449208548}{75943352179}a^{8}+\frac{22751663832631}{75943352179}a^{7}-\frac{7414340839363}{75943352179}a^{6}-\frac{40270576324837}{75943352179}a^{5}+\frac{4540146566702}{75943352179}a^{4}+\frac{32769956097206}{75943352179}a^{3}+\frac{295434884091}{75943352179}a^{2}-\frac{6375387695292}{75943352179}a+\frac{141644237656}{75943352179}$, $\frac{5}{75943352179}a^{27}-\frac{2}{75943352179}a^{26}-\frac{903}{75943352179}a^{25}+\frac{387}{75943352179}a^{24}+\frac{72695}{75943352179}a^{23}-\frac{32577}{75943352179}a^{22}-\frac{3443361}{75943352179}a^{21}+\frac{1580687}{75943352179}a^{20}+\frac{106655302}{75943352179}a^{19}-\frac{49235174}{75943352179}a^{18}-\frac{2270080117}{75943352179}a^{17}+\frac{1035602036}{75943352179}a^{16}+\frac{33944307772}{75943352179}a^{15}-\frac{15046865801}{75943352179}a^{14}-\frac{358072715780}{75943352179}a^{13}+\frac{151744895834}{75943352179}a^{12}+\frac{2634905191501}{75943352179}a^{11}-\frac{1051857625456}{75943352179}a^{10}-\frac{13143711323831}{75943352179}a^{9}+\frac{4888018641093}{75943352179}a^{8}+\frac{42122613893806}{75943352179}a^{7}-\frac{14543551060943}{75943352179}a^{6}-\frac{78494637638239}{75943352179}a^{5}+\frac{25426571918377}{75943352179}a^{4}+\frac{68620906779322}{75943352179}a^{3}-\frac{21525667227395}{75943352179}a^{2}-\frac{13306423269132}{75943352179}a+\frac{3992201828556}{75943352179}$, $\frac{3}{75943352179}a^{27}+\frac{15}{75943352179}a^{26}-\frac{519}{75943352179}a^{25}-\frac{2397}{75943352179}a^{24}+\frac{40311}{75943352179}a^{23}+\frac{170236}{75943352179}a^{22}-\frac{1855448}{75943352179}a^{21}-\frac{7077465}{75943352179}a^{20}+\frac{56247267}{75943352179}a^{19}+\frac{190900095}{75943352179}a^{18}-\frac{1180022559}{75943352179}a^{17}-\frac{3497265352}{75943352179}a^{16}+\frac{17511953175}{75943352179}a^{15}+\frac{44246589206}{75943352179}a^{14}-\frac{184521830282}{75943352179}a^{13}-\frac{384986510433}{75943352179}a^{12}+\frac{1363852214799}{75943352179}a^{11}+\frac{2246957886327}{75943352179}a^{10}-\frac{6861689485431}{75943352179}a^{9}-\frac{8339828091197}{75943352179}a^{8}+\frac{22222392071629}{75943352179}a^{7}+\frac{17702522769302}{75943352179}a^{6}-\frac{41847191910219}{75943352179}a^{5}-\frac{16674878487153}{75943352179}a^{4}+\frac{37181300933829}{75943352179}a^{3}+\frac{1141710638919}{75943352179}a^{2}-\frac{8187781313898}{75943352179}a+\frac{1211357178623}{75943352179}$, $\frac{7}{75943352179}a^{27}-\frac{1104}{75943352179}a^{25}+\frac{77092}{75943352179}a^{23}-\frac{3139713}{75943352179}a^{21}+\frac{82705692}{75943352179}a^{19}-\frac{1477549525}{75943352179}a^{17}+\frac{18267428478}{75943352179}a^{15}-\frac{156758745276}{75943352179}a^{13}-\frac{1609901}{75943352179}a^{12}+\frac{923695469424}{75943352179}a^{11}+\frac{136710253}{75943352179}a^{10}-\frac{3645745172784}{75943352179}a^{9}-\frac{4306109949}{75943352179}a^{8}+\frac{9222001398720}{75943352179}a^{7}+\frac{60827879725}{75943352179}a^{6}-\frac{13794395354496}{75943352179}a^{5}-\frac{364384257407}{75943352179}a^{4}+\frac{10319597655552}{75943352179}a^{3}+\frac{686007252820}{75943352179}a^{2}-\frac{2494117225728}{75943352179}a+\frac{164754385156}{75943352179}$ 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:  \( 260881845825591150000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{28}\cdot(2\pi)^{0}\cdot 260881845825591150000 \cdot 1}{2\cdot\sqrt{35394489068231220324814698212289719250778220848093751207381}}\cr\approx \mathstrut & 0.186116946081246 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 - x^27 - 173*x^26 + 173*x^25 + 13399*x^24 - 13399*x^23 - 613001*x^22 + 613001*x^21 + 18404503*x^20 - 18404503*x^19 - 380963081*x^18 + 380963081*x^17 + 5557459255*x^16 - 5557459255*x^15 - 57374393033*x^14 + 57374393033*x^13 + 414614499127*x^12 - 414614499127*x^11 - 2039727740105*x^10 + 2039727740105*x^9 + 6485882143543*x^8 - 6485882143543*x^7 - 12115448511689*x^6 + 12115448511689*x^5 + 10862665827127*x^4 - 10862665827127*x^3 - 2393938599113*x^2 + 2393938599113*x - 121377840329)
 
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 - 173*x^26 + 173*x^25 + 13399*x^24 - 13399*x^23 - 613001*x^22 + 613001*x^21 + 18404503*x^20 - 18404503*x^19 - 380963081*x^18 + 380963081*x^17 + 5557459255*x^16 - 5557459255*x^15 - 57374393033*x^14 + 57374393033*x^13 + 414614499127*x^12 - 414614499127*x^11 - 2039727740105*x^10 + 2039727740105*x^9 + 6485882143543*x^8 - 6485882143543*x^7 - 12115448511689*x^6 + 12115448511689*x^5 + 10862665827127*x^4 - 10862665827127*x^3 - 2393938599113*x^2 + 2393938599113*x - 121377840329, 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 - 173*x^26 + 173*x^25 + 13399*x^24 - 13399*x^23 - 613001*x^22 + 613001*x^21 + 18404503*x^20 - 18404503*x^19 - 380963081*x^18 + 380963081*x^17 + 5557459255*x^16 - 5557459255*x^15 - 57374393033*x^14 + 57374393033*x^13 + 414614499127*x^12 - 414614499127*x^11 - 2039727740105*x^10 + 2039727740105*x^9 + 6485882143543*x^8 - 6485882143543*x^7 - 12115448511689*x^6 + 12115448511689*x^5 + 10862665827127*x^4 - 10862665827127*x^3 - 2393938599113*x^2 + 2393938599113*x - 121377840329);
 
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 - 173*x^26 + 173*x^25 + 13399*x^24 - 13399*x^23 - 613001*x^22 + 613001*x^21 + 18404503*x^20 - 18404503*x^19 - 380963081*x^18 + 380963081*x^17 + 5557459255*x^16 - 5557459255*x^15 - 57374393033*x^14 + 57374393033*x^13 + 414614499127*x^12 - 414614499127*x^11 - 2039727740105*x^10 + 2039727740105*x^9 + 6485882143543*x^8 - 6485882143543*x^7 - 12115448511689*x^6 + 12115448511689*x^5 + 10862665827127*x^4 - 10862665827127*x^3 - 2393938599113*x^2 + 2393938599113*x - 121377840329);
 
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 $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.12901781.1, 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 $28$ $28$ ${\href{/padicField/5.7.0.1}{7} }^{4}$ ${\href{/padicField/7.14.0.1}{14} }^{2}$ $28$ ${\href{/padicField/13.14.0.1}{14} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }^{7}$ $28$ R 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
\(23\) Copy content Toggle raw display 23.14.7.2$x^{14} - 21574 x^{13} + 234933293 x^{12} - 1215548742590 x^{11} + 4112603302919993 x^{10} - 2725136947640868418 x^{9} + 363970304488058959670 x^{8} + 412439955621146008597774 x^{7} + 8371317003225356072410 x^{6} - 1441597445302019393122 x^{5} + 50038044386627554831 x^{4} - 340160375675128190 x^{3} + 1512111255867499 x^{2} - 3193726269286 x + 3404825447$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
23.14.7.2$x^{14} - 21574 x^{13} + 234933293 x^{12} - 1215548742590 x^{11} + 4112603302919993 x^{10} - 2725136947640868418 x^{9} + 363970304488058959670 x^{8} + 412439955621146008597774 x^{7} + 8371317003225356072410 x^{6} - 1441597445302019393122 x^{5} + 50038044386627554831 x^{4} - 340160375675128190 x^{3} + 1512111255867499 x^{2} - 3193726269286 x + 3404825447$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
\(29\) Copy content Toggle raw display Deg $28$$28$$1$$27$