↑

Properties

Label 23.11.110...432.1
Degree $23$
Signature $[11, 6]$
Discriminant $1.106\times 10^{42}$
Root discriminant \(67.30\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{23}$ (as 23T7)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 18*x^3 + 39*x^2 - 2*x - 1)
 
Copy content gp:K = bnfinit(y^23 - 7*y^22 + 2*y^21 + 88*y^20 - 150*y^19 - 422*y^18 + 1148*y^17 + 862*y^16 - 4167*y^15 - 105*y^14 + 8563*y^13 - 2935*y^12 - 10393*y^11 + 5920*y^10 + 7285*y^9 - 5405*y^8 - 2661*y^7 + 2503*y^6 + 354*y^5 - 537*y^4 + 18*y^3 + 39*y^2 - 2*y - 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 18*x^3 + 39*x^2 - 2*x - 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 18*x^3 + 39*x^2 - 2*x - 1)
 

\( x^{23} - 7 x^{22} + 2 x^{21} + 88 x^{20} - 150 x^{19} - 422 x^{18} + 1148 x^{17} + 862 x^{16} - 4167 x^{15} + \cdots - 1 \) 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:  $23$
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:  $[11, 6]$
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:   \(1105964546020085080320539114269649147026432\) \(\medspace = 2^{10}\cdot 22483\cdot 2527336563667297\cdot 19007452217635529293\) 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:  \(67.30\)
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:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(22483\), \(2527336563667297\), \(19007452217635529293\) 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:  $\Q(\sqrt{10800\!\cdots\!45143}$)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$ 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:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
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:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

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:  $16$
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:   $a$, $a^{21}-7a^{20}+2a^{19}+88a^{18}-150a^{17}-422a^{16}+1148a^{15}+862a^{14}-4167a^{13}-105a^{12}+8562a^{11}-2932a^{10}-10385a^{9}+5893a^{8}+7263a^{7}-5316a^{6}-2638a^{5}+2371a^{4}+349a^{3}-450a^{2}+14a+18$, $a^{21}-6a^{20}-4a^{19}+84a^{18}-66a^{17}-488a^{16}+660a^{15}+1522a^{14}-2645a^{13}-2750a^{12}+5813a^{11}+2878a^{10}-7514a^{9}-1597a^{8}+5682a^{7}+298a^{6}-2351a^{5}+101a^{4}+445a^{3}-41a^{2}-20a+1$, $5a^{22}-35a^{21}+11a^{20}+434a^{19}-753a^{18}-2031a^{17}+5667a^{16}+3888a^{15}-20183a^{14}+635a^{13}+40381a^{12}-16350a^{11}-46963a^{10}+30588a^{9}+30388a^{8}-26604a^{7}-9056a^{6}+11532a^{5}+148a^{4}-2135a^{3}+363a^{2}+79a-18$, $a^{22}-7a^{21}+2a^{20}+88a^{19}-150a^{18}-422a^{17}+1148a^{16}+862a^{15}-4167a^{14}-105a^{13}+8563a^{12}-2935a^{11}-10393a^{10}+5920a^{9}+7285a^{8}-5405a^{7}-2661a^{6}+2502a^{5}+356a^{4}-533a^{3}+11a^{2}+35a+2$, $4a^{22}-30a^{21}+22a^{20}+347a^{19}-771a^{18}-1380a^{17}+5365a^{16}+1155a^{15}-17972a^{14}+7677a^{13}+33123a^{12}-27796a^{11}-33225a^{10}+42192a^{9}+14669a^{8}-33561a^{7}+1567a^{6}+13673a^{5}-3713a^{4}-2312a^{3}+952a^{2}+35a-34$, $5a^{22}-31a^{21}-16a^{20}+436a^{19}-406a^{18}-2541a^{17}+3916a^{16}+7915a^{15}-16010a^{14}-14108a^{13}+36766a^{12}+14047a^{11}-50988a^{10}-6418a^{9}+43015a^{8}-440a^{7}-21235a^{6}+1640a^{5}+5556a^{4}-535a^{3}-627a^{2}+33a+19$, $8a^{22}-54a^{21}+3a^{20}+701a^{19}-1022a^{18}-3588a^{17}+8190a^{16}+8772a^{15}-30474a^{14}-8320a^{13}+64234a^{12}-6520a^{11}-80892a^{10}+24122a^{9}+60595a^{8}-24047a^{7}-25606a^{6}+10971a^{5}+5379a^{4}-2143a^{3}-452a^{2}+102a+18$, $5a^{22}-36a^{21}+17a^{20}+438a^{19}-838a^{18}-1959a^{17}+6157a^{16}+3156a^{15}-21635a^{14}+3624a^{13}+42610a^{12}-22993a^{11}-48232a^{10}+39153a^{9}+29391a^{8}-32919a^{7}-7104a^{6}+13969a^{5}-940a^{4}-2509a^{3}+584a^{2}+78a-26$, $a^{22}-7a^{21}+2a^{20}+88a^{19}-150a^{18}-422a^{17}+1148a^{16}+861a^{15}-4162a^{14}-102a^{13}+8516a^{12}-2908a^{11}-10225a^{10}+5760a^{9}+6996a^{8}-5080a^{7}-2408a^{6}+2197a^{5}+244a^{4}-399a^{3}+36a^{2}+12a-1$, $12a^{22}-79a^{21}-9a^{20}+1054a^{19}-1369a^{18}-5638a^{17}+11525a^{16}+15026a^{15}-44161a^{14}-18781a^{13}+95664a^{12}+2014a^{11}-124054a^{10}+23351a^{9}+96052a^{8}-28011a^{7}-42194a^{6}+13661a^{5}+9318a^{4}-2728a^{3}-840a^{2}+123a+27$, $38a^{22}-253a^{21}-11a^{20}+3343a^{19}-4555a^{18}-17633a^{17}+37631a^{16}+45849a^{15}-143037a^{14}-53554a^{13}+308599a^{12}-5053a^{11}-400094a^{10}+87738a^{9}+311393a^{8}-99613a^{7}-138708a^{6}+48796a^{5}+31605a^{4}-10210a^{3}-3072a^{2}+562a+126$, $18a^{22}-120a^{21}-4a^{20}+1583a^{19}-2174a^{18}-8323a^{17}+17912a^{16}+21483a^{15}-67970a^{14}-24463a^{13}+146362a^{12}-4386a^{11}-189240a^{10}+44183a^{9}+146668a^{8}-49220a^{7}-64887a^{6}+23976a^{5}+14615a^{4}-5000a^{3}-1400a^{2}+276a+58$, $a^{22}-11a^{21}+29a^{20}+86a^{19}-498a^{18}+94a^{17}+2902a^{16}-3241a^{15}-8279a^{14}+15032a^{13}+11676a^{12}-34409a^{11}-4704a^{10}+44585a^{9}-8250a^{8}-32967a^{7}+12315a^{6}+12913a^{5}-6469a^{4}-2112a^{3}+1330a^{2}+37a-55$, $a^{21}-6a^{20}-4a^{19}+84a^{18}-66a^{17}-487a^{16}+655a^{15}+1517a^{14}-2588a^{13}-2771a^{12}+5553a^{11}+3085a^{10}-6903a^{9}-2181a^{8}+4888a^{7}+1050a^{6}-1790a^{5}-324a^{4}+262a^{3}+28a^{2}-11a-1$, $3a^{22}-21a^{21}+6a^{20}+264a^{19}-450a^{18}-1266a^{17}+3443a^{16}+2590a^{15}-12491a^{14}-368a^{13}+25658a^{12}-8520a^{11}-31177a^{10}+16962a^{9}+22031a^{8}-14973a^{7}-8345a^{6}+6448a^{5}+1348a^{4}-1158a^{3}-32a^{2}+45a+2$ Copy content Toggle raw display (assuming GRH)
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:  \( 11886760766000 \) (assuming GRH)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{6}\cdot 11886760766000 \cdot 1}{2\cdot\sqrt{1105964546020085080320539114269649147026432}}\cr\approx \mathstrut & 0.712150996690076 \end{aligned}\] (assuming GRH)

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^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 18*x^3 + 39*x^2 - 2*x - 1) 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^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 18*x^3 + 39*x^2 - 2*x - 1, 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^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 18*x^3 + 39*x^2 - 2*x - 1); 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 = PolynomialRing(QQ); K, a = NumberField(x^23 - 7*x^22 + 2*x^21 + 88*x^20 - 150*x^19 - 422*x^18 + 1148*x^17 + 862*x^16 - 4167*x^15 - 105*x^14 + 8563*x^13 - 2935*x^12 - 10393*x^11 + 5920*x^10 + 7285*x^9 - 5405*x^8 - 2661*x^7 + 2503*x^6 + 354*x^5 - 537*x^4 + 18*x^3 + 39*x^2 - 2*x - 1); 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

$S_{23}$ (as 23T7):

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); G, transitive_group_identification(G)
 
A non-solvable group of order 25852016738884976640000
The 1255 conjugacy class representatives for $S_{23}$
Character table for $S_{23}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 46 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ $22{,}\,{\href{/padicField/5.1.0.1}{1} }$ $22{,}\,{\href{/padicField/7.1.0.1}{1} }$ $22{,}\,{\href{/padicField/11.1.0.1}{1} }$ $15{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ $21{,}\,{\href{/padicField/17.2.0.1}{2} }$ $22{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ $15{,}\,{\href{/padicField/29.8.0.1}{8} }$ ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ $23$ $17{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ $18{,}\,{\href{/padicField/47.5.0.1}{5} }$ $15{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.1.4.4a1.1$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$$
2.3.2.6a2.1$x^{6} + 4 x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 3$$2$$3$$6$$A_4\times C_2$$$[2, 2, 2]^{3}$$
2.13.1.0a1.1$x^{13} + x^{4} + x^{3} + x + 1$$1$$13$$0$$C_{13}$$$[\ ]^{13}$$
\(22483\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $3$$1$$3$$0$$C_3$$$[\ ]^{3}$$
Deg $16$$1$$16$$0$$C_{16}$$$[\ ]^{16}$$
\(2527336563667297\) Copy content Toggle raw display $\Q_{2527336563667297}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $20$$1$$20$$0$20T1$$[\ ]^{20}$$
\(19007452217635529293\) Copy content Toggle raw display $\Q_{19007452217635529293}$$x$$1$$1$$0$Trivial$$[\ ]$$
Deg $2$$2$$1$$1$$C_2$$$[\ ]_{2}$$
Deg $5$$1$$5$$0$$C_5$$$[\ ]^{5}$$
Deg $15$$1$$15$$0$$C_{15}$$$[\ ]^{15}$$

Spectrum of ring of integers

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