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Properties

Label 28.0.482...000.5
Degree $28$
Signature $[0, 14]$
Discriminant $4.828\times 10^{64}$
Root discriminant \(204.24\)
Ramified primes $2,5,7$
Class number not computed
Class group not computed
Galois group $C_2\times C_{14}$ (as 28T2)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^28 - 70*x^26 - 84*x^25 + 2422*x^24 + 5068*x^23 - 38626*x^22 - 104408*x^21 + 430633*x^20 + 1651972*x^19 + 182000*x^18 - 7617288*x^17 - 6150963*x^16 + 59290000*x^15 + 448416620*x^14 + 1054129104*x^13 + 2845635177*x^12 + 4291096796*x^11 + 20891101756*x^10 + 52970184596*x^9 + 304787155988*x^8 + 674144620360*x^7 + 2487436552174*x^6 + 3848216248772*x^5 + 10252279894346*x^4 + 10477877455052*x^3 + 21520460035676*x^2 + 11679956181964*x + 18055489522361)
 
Copy content gp:K = bnfinit(y^28 - 70*y^26 - 84*y^25 + 2422*y^24 + 5068*y^23 - 38626*y^22 - 104408*y^21 + 430633*y^20 + 1651972*y^19 + 182000*y^18 - 7617288*y^17 - 6150963*y^16 + 59290000*y^15 + 448416620*y^14 + 1054129104*y^13 + 2845635177*y^12 + 4291096796*y^11 + 20891101756*y^10 + 52970184596*y^9 + 304787155988*y^8 + 674144620360*y^7 + 2487436552174*y^6 + 3848216248772*y^5 + 10252279894346*y^4 + 10477877455052*y^3 + 21520460035676*y^2 + 11679956181964*y + 18055489522361, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 70*x^26 - 84*x^25 + 2422*x^24 + 5068*x^23 - 38626*x^22 - 104408*x^21 + 430633*x^20 + 1651972*x^19 + 182000*x^18 - 7617288*x^17 - 6150963*x^16 + 59290000*x^15 + 448416620*x^14 + 1054129104*x^13 + 2845635177*x^12 + 4291096796*x^11 + 20891101756*x^10 + 52970184596*x^9 + 304787155988*x^8 + 674144620360*x^7 + 2487436552174*x^6 + 3848216248772*x^5 + 10252279894346*x^4 + 10477877455052*x^3 + 21520460035676*x^2 + 11679956181964*x + 18055489522361);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^28 - 70*x^26 - 84*x^25 + 2422*x^24 + 5068*x^23 - 38626*x^22 - 104408*x^21 + 430633*x^20 + 1651972*x^19 + 182000*x^18 - 7617288*x^17 - 6150963*x^16 + 59290000*x^15 + 448416620*x^14 + 1054129104*x^13 + 2845635177*x^12 + 4291096796*x^11 + 20891101756*x^10 + 52970184596*x^9 + 304787155988*x^8 + 674144620360*x^7 + 2487436552174*x^6 + 3848216248772*x^5 + 10252279894346*x^4 + 10477877455052*x^3 + 21520460035676*x^2 + 11679956181964*x + 18055489522361)
 

\( x^{28} - 70 x^{26} - 84 x^{25} + 2422 x^{24} + 5068 x^{23} - 38626 x^{22} - 104408 x^{21} + \cdots + 18055489522361 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $28$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[0, 14]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(48277178382311752679795381206864912282636228388454400000000000000\) \(\medspace = 2^{42}\cdot 5^{14}\cdot 7^{50}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(204.24\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{3/2}5^{1/2}7^{25/14}\approx 204.23663129490222$
Ramified primes:   \(2\), \(5\), \(7\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_2\times C_{14}$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1960=2^{3}\cdot 5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1960}(1,·)$, $\chi_{1960}(1861,·)$, $\chi_{1960}(769,·)$, $\chi_{1960}(841,·)$, $\chi_{1960}(209,·)$, $\chi_{1960}(461,·)$, $\chi_{1960}(1581,·)$, $\chi_{1960}(1681,·)$, $\chi_{1960}(1429,·)$, $\chi_{1960}(1049,·)$, $\chi_{1960}(281,·)$, $\chi_{1960}(29,·)$, $\chi_{1960}(589,·)$, $\chi_{1960}(741,·)$, $\chi_{1960}(1889,·)$, $\chi_{1960}(1301,·)$, $\chi_{1960}(869,·)$, $\chi_{1960}(1329,·)$, $\chi_{1960}(489,·)$, $\chi_{1960}(1709,·)$, $\chi_{1960}(1021,·)$, $\chi_{1960}(561,·)$, $\chi_{1960}(309,·)$, $\chi_{1960}(1609,·)$, $\chi_{1960}(1401,·)$, $\chi_{1960}(1121,·)$, $\chi_{1960}(1149,·)$, $\chi_{1960}(181,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{8192}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{98}a^{14}+\frac{1}{7}a^{12}-\frac{3}{7}a^{11}+\frac{5}{14}a^{10}-\frac{1}{7}a^{9}-\frac{1}{14}a^{8}+\frac{15}{49}a^{7}+\frac{5}{14}a^{6}+\frac{1}{7}a^{5}+\frac{3}{7}a^{4}-\frac{3}{7}a^{3}-\frac{3}{7}a^{2}+\frac{1}{7}a-\frac{19}{98}$, $\frac{1}{98}a^{15}+\frac{1}{7}a^{13}-\frac{3}{7}a^{12}+\frac{5}{14}a^{11}-\frac{1}{7}a^{10}-\frac{1}{14}a^{9}+\frac{15}{49}a^{8}+\frac{5}{14}a^{7}+\frac{1}{7}a^{6}+\frac{3}{7}a^{5}-\frac{3}{7}a^{4}-\frac{3}{7}a^{3}+\frac{1}{7}a^{2}-\frac{19}{98}a$, $\frac{1}{98}a^{16}-\frac{3}{7}a^{13}+\frac{5}{14}a^{12}-\frac{1}{7}a^{11}-\frac{1}{14}a^{10}+\frac{15}{49}a^{9}+\frac{5}{14}a^{8}-\frac{1}{7}a^{7}+\frac{3}{7}a^{6}-\frac{3}{7}a^{5}-\frac{3}{7}a^{4}+\frac{1}{7}a^{3}-\frac{19}{98}a^{2}-\frac{2}{7}$, $\frac{1}{98}a^{17}+\frac{5}{14}a^{13}-\frac{1}{7}a^{12}-\frac{1}{14}a^{11}+\frac{15}{49}a^{10}+\frac{5}{14}a^{9}-\frac{1}{7}a^{8}+\frac{2}{7}a^{7}-\frac{3}{7}a^{6}-\frac{3}{7}a^{5}+\frac{1}{7}a^{4}-\frac{19}{98}a^{3}-\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{98}a^{18}-\frac{1}{7}a^{13}-\frac{1}{14}a^{12}+\frac{15}{49}a^{11}-\frac{1}{7}a^{10}-\frac{1}{7}a^{9}-\frac{3}{14}a^{8}-\frac{1}{7}a^{7}+\frac{1}{14}a^{6}+\frac{1}{7}a^{5}-\frac{19}{98}a^{4}-\frac{2}{7}a^{2}-\frac{1}{7}a-\frac{3}{14}$, $\frac{1}{98}a^{19}-\frac{1}{14}a^{13}+\frac{15}{49}a^{12}-\frac{1}{7}a^{11}-\frac{1}{7}a^{10}-\frac{3}{14}a^{9}-\frac{1}{7}a^{8}+\frac{5}{14}a^{7}+\frac{1}{7}a^{6}-\frac{19}{98}a^{5}-\frac{2}{7}a^{3}-\frac{1}{7}a^{2}-\frac{3}{14}a+\frac{2}{7}$, $\frac{1}{98}a^{20}+\frac{15}{49}a^{13}-\frac{1}{7}a^{12}-\frac{1}{7}a^{11}+\frac{2}{7}a^{10}-\frac{1}{7}a^{9}-\frac{1}{7}a^{8}+\frac{2}{7}a^{7}+\frac{15}{49}a^{6}-\frac{2}{7}a^{4}-\frac{1}{7}a^{3}-\frac{3}{14}a^{2}+\frac{2}{7}a-\frac{5}{14}$, $\frac{1}{294}a^{21}-\frac{1}{294}a^{19}+\frac{13}{42}a^{13}-\frac{12}{49}a^{12}-\frac{5}{21}a^{11}+\frac{3}{7}a^{10}+\frac{5}{42}a^{9}-\frac{1}{7}a^{8}+\frac{25}{98}a^{7}+\frac{1}{21}a^{6}+\frac{61}{294}a^{5}+\frac{1}{3}a^{4}+\frac{13}{42}a^{3}-\frac{5}{21}a^{2}+\frac{4}{21}a+\frac{26}{147}$, $\frac{1}{173166}a^{22}-\frac{11}{24738}a^{21}+\frac{1}{4557}a^{20}-\frac{193}{173166}a^{19}+\frac{251}{57722}a^{18}+\frac{11}{28861}a^{17}-\frac{277}{57722}a^{16}+\frac{131}{28861}a^{15}-\frac{23}{9114}a^{14}+\frac{49531}{173166}a^{13}+\frac{3055}{86583}a^{12}-\frac{36662}{86583}a^{11}+\frac{23971}{86583}a^{10}-\frac{85921}{173166}a^{9}+\frac{25301}{57722}a^{8}-\frac{2125}{9114}a^{7}+\frac{4411}{28861}a^{6}-\frac{12633}{57722}a^{5}+\frac{11045}{28861}a^{4}+\frac{597}{57722}a^{3}-\frac{23602}{86583}a^{2}-\frac{1138}{4123}a+\frac{14312}{86583}$, $\frac{1}{173166}a^{23}-\frac{1}{173166}a^{21}-\frac{267}{57722}a^{20}-\frac{187}{57722}a^{19}-\frac{44}{28861}a^{18}+\frac{239}{57722}a^{17}+\frac{137}{57722}a^{16}+\frac{1}{24738}a^{15}-\frac{1}{8246}a^{14}+\frac{40807}{86583}a^{13}+\frac{16349}{57722}a^{12}-\frac{33059}{86583}a^{11}+\frac{3984}{28861}a^{10}+\frac{4989}{57722}a^{9}+\frac{3124}{12369}a^{8}+\frac{18173}{86583}a^{7}-\frac{27355}{173166}a^{6}+\frac{28559}{86583}a^{5}-\frac{18989}{86583}a^{4}-\frac{48301}{173166}a^{3}-\frac{1552}{4557}a^{2}+\frac{3961}{8246}a+\frac{4910}{28861}$, $\frac{1}{173166}a^{24}-\frac{289}{173166}a^{21}-\frac{523}{173166}a^{20}+\frac{103}{24738}a^{19}-\frac{99}{57722}a^{18}+\frac{159}{57722}a^{17}-\frac{412}{86583}a^{16}+\frac{255}{57722}a^{15}-\frac{5}{8246}a^{14}-\frac{4310}{86583}a^{13}+\frac{5311}{24738}a^{12}-\frac{22354}{86583}a^{11}+\frac{473}{1302}a^{10}+\frac{21671}{57722}a^{9}-\frac{11905}{86583}a^{8}+\frac{24446}{86583}a^{7}+\frac{9995}{57722}a^{6}-\frac{1145}{8246}a^{5}-\frac{7213}{86583}a^{4}+\frac{35314}{86583}a^{3}-\frac{9997}{57722}a^{2}+\frac{76493}{173166}a-\frac{41467}{173166}$, $\frac{1}{13680114}a^{25}-\frac{1}{6840057}a^{24}-\frac{2}{977151}a^{23}-\frac{2}{977151}a^{22}+\frac{1184}{6840057}a^{21}-\frac{6299}{4560038}a^{20}+\frac{244}{51429}a^{19}+\frac{22979}{4560038}a^{18}+\frac{46615}{13680114}a^{17}+\frac{103}{720006}a^{16}+\frac{10657}{6840057}a^{15}-\frac{1723}{13680114}a^{14}+\frac{733497}{4560038}a^{13}+\frac{399331}{977151}a^{12}-\frac{17491}{1954302}a^{11}+\frac{555916}{6840057}a^{10}+\frac{4240601}{13680114}a^{9}+\frac{1719323}{4560038}a^{8}+\frac{1552003}{4560038}a^{7}+\frac{2197702}{6840057}a^{6}-\frac{52987}{279186}a^{5}-\frac{4518145}{13680114}a^{4}+\frac{2835467}{13680114}a^{3}+\frac{66946}{360003}a^{2}+\frac{147886}{360003}a-\frac{5435699}{13680114}$, $\frac{1}{37\cdots 46}a^{26}-\frac{11\cdots 68}{62\cdots 41}a^{25}+\frac{35\cdots 99}{37\cdots 46}a^{24}-\frac{64\cdots 97}{28\cdots 62}a^{23}+\frac{73\cdots 93}{37\cdots 46}a^{22}+\frac{64\cdots 14}{62\cdots 41}a^{21}-\frac{25\cdots 32}{62\cdots 41}a^{20}-\frac{50\cdots 39}{37\cdots 46}a^{19}+\frac{74\cdots 23}{18\cdots 23}a^{18}-\frac{50\cdots 39}{12\cdots 82}a^{17}-\frac{43\cdots 03}{37\cdots 46}a^{16}+\frac{59\cdots 01}{18\cdots 23}a^{15}+\frac{26\cdots 28}{62\cdots 41}a^{14}-\frac{26\cdots 32}{62\cdots 41}a^{13}+\frac{18\cdots 43}{37\cdots 46}a^{12}+\frac{17\cdots 87}{12\cdots 82}a^{11}+\frac{93\cdots 25}{37\cdots 46}a^{10}+\frac{26\cdots 68}{18\cdots 23}a^{9}-\frac{88\cdots 11}{18\cdots 23}a^{8}+\frac{14\cdots 07}{37\cdots 46}a^{7}-\frac{48\cdots 34}{98\cdots 17}a^{6}+\frac{29\cdots 22}{18\cdots 23}a^{5}-\frac{14\cdots 28}{18\cdots 23}a^{4}+\frac{53\cdots 40}{26\cdots 89}a^{3}+\frac{12\cdots 03}{62\cdots 41}a^{2}-\frac{43\cdots 99}{12\cdots 66}a+\frac{71\cdots 35}{18\cdots 23}$, $\frac{1}{76\cdots 86}a^{27}-\frac{64\cdots 03}{25\cdots 62}a^{26}+\frac{25\cdots 30}{12\cdots 31}a^{25}-\frac{46\cdots 45}{25\cdots 62}a^{24}+\frac{91\cdots 71}{38\cdots 93}a^{23}+\frac{91\cdots 86}{38\cdots 93}a^{22}+\frac{79\cdots 11}{82\cdots 02}a^{21}-\frac{11\cdots 41}{76\cdots 86}a^{20}+\frac{20\cdots 93}{76\cdots 86}a^{19}-\frac{75\cdots 25}{51\cdots 38}a^{18}-\frac{31\cdots 45}{25\cdots 62}a^{17}+\frac{91\cdots 07}{25\cdots 62}a^{16}+\frac{46\cdots 71}{96\cdots 34}a^{15}-\frac{29\cdots 15}{10\cdots 98}a^{14}-\frac{25\cdots 06}{54\cdots 99}a^{13}+\frac{49\cdots 17}{12\cdots 31}a^{12}+\frac{14\cdots 19}{38\cdots 93}a^{11}+\frac{13\cdots 27}{38\cdots 93}a^{10}-\frac{11\cdots 31}{54\cdots 99}a^{9}-\frac{29\cdots 41}{38\cdots 93}a^{8}-\frac{41\cdots 49}{76\cdots 86}a^{7}-\frac{15\cdots 72}{38\cdots 93}a^{6}+\frac{23\cdots 33}{13\cdots 98}a^{5}+\frac{32\cdots 74}{12\cdots 31}a^{4}-\frac{12\cdots 20}{38\cdots 93}a^{3}-\frac{11\cdots 73}{38\cdots 93}a^{2}-\frac{95\cdots 83}{76\cdots 86}a+\frac{66\cdots 77}{25\cdots 62}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  not computed
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  not computed
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 
Relative class number:   data not computed

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $13$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:  not computed
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot R \cdot h}{2\cdot\sqrt{48277178382311752679795381206864912282636228388454400000000000000}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^28 - 70*x^26 - 84*x^25 + 2422*x^24 + 5068*x^23 - 38626*x^22 - 104408*x^21 + 430633*x^20 + 1651972*x^19 + 182000*x^18 - 7617288*x^17 - 6150963*x^16 + 59290000*x^15 + 448416620*x^14 + 1054129104*x^13 + 2845635177*x^12 + 4291096796*x^11 + 20891101756*x^10 + 52970184596*x^9 + 304787155988*x^8 + 674144620360*x^7 + 2487436552174*x^6 + 3848216248772*x^5 + 10252279894346*x^4 + 10477877455052*x^3 + 21520460035676*x^2 + 11679956181964*x + 18055489522361) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^28 - 70*x^26 - 84*x^25 + 2422*x^24 + 5068*x^23 - 38626*x^22 - 104408*x^21 + 430633*x^20 + 1651972*x^19 + 182000*x^18 - 7617288*x^17 - 6150963*x^16 + 59290000*x^15 + 448416620*x^14 + 1054129104*x^13 + 2845635177*x^12 + 4291096796*x^11 + 20891101756*x^10 + 52970184596*x^9 + 304787155988*x^8 + 674144620360*x^7 + 2487436552174*x^6 + 3848216248772*x^5 + 10252279894346*x^4 + 10477877455052*x^3 + 21520460035676*x^2 + 11679956181964*x + 18055489522361, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 70*x^26 - 84*x^25 + 2422*x^24 + 5068*x^23 - 38626*x^22 - 104408*x^21 + 430633*x^20 + 1651972*x^19 + 182000*x^18 - 7617288*x^17 - 6150963*x^16 + 59290000*x^15 + 448416620*x^14 + 1054129104*x^13 + 2845635177*x^12 + 4291096796*x^11 + 20891101756*x^10 + 52970184596*x^9 + 304787155988*x^8 + 674144620360*x^7 + 2487436552174*x^6 + 3848216248772*x^5 + 10252279894346*x^4 + 10477877455052*x^3 + 21520460035676*x^2 + 11679956181964*x + 18055489522361); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^28 - 70*x^26 - 84*x^25 + 2422*x^24 + 5068*x^23 - 38626*x^22 - 104408*x^21 + 430633*x^20 + 1651972*x^19 + 182000*x^18 - 7617288*x^17 - 6150963*x^16 + 59290000*x^15 + 448416620*x^14 + 1054129104*x^13 + 2845635177*x^12 + 4291096796*x^11 + 20891101756*x^10 + 52970184596*x^9 + 304787155988*x^8 + 674144620360*x^7 + 2487436552174*x^6 + 3848216248772*x^5 + 10252279894346*x^4 + 10477877455052*x^3 + 21520460035676*x^2 + 11679956181964*x + 18055489522361); 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

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

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$

Intermediate fields

\(\Q(\sqrt{-14}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{10}, \sqrt{-14})\), 7.7.13841287201.1, 14.0.2812424737865523319657201664.1, 14.14.31388668949392001335459840000000.1, 14.0.104770985911247257875546875.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content 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 R ${\href{/padicField/3.7.0.1}{7} }^{4}$ R R ${\href{/padicField/11.14.0.1}{14} }^{2}$ ${\href{/padicField/13.7.0.1}{7} }^{4}$ ${\href{/padicField/17.14.0.1}{14} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{14}$ ${\href{/padicField/23.14.0.1}{14} }^{2}$ ${\href{/padicField/29.14.0.1}{14} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{14}$ ${\href{/padicField/37.14.0.1}{14} }^{2}$ ${\href{/padicField/41.14.0.1}{14} }^{2}$ ${\href{/padicField/43.14.0.1}{14} }^{2}$ ${\href{/padicField/47.14.0.1}{14} }^{2}$ ${\href{/padicField/53.14.0.1}{14} }^{2}$ ${\href{/padicField/59.14.0.1}{14} }^{2}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
\(2\) Copy content Toggle raw display Deg $28$$2$$14$$42$
\(5\) Copy content Toggle raw display 5.7.2.7a1.1$x^{14} + 6 x^{8} + 6 x^{7} + 9 x^{2} + 23 x + 9$$2$$7$$7$$C_{14}$$$[\ ]_{2}^{7}$$
5.7.2.7a1.1$x^{14} + 6 x^{8} + 6 x^{7} + 9 x^{2} + 23 x + 9$$2$$7$$7$$C_{14}$$$[\ ]_{2}^{7}$$
\(7\) Copy content Toggle raw display Deg $28$$14$$2$$50$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)