Properties

Label 21.21.122...944.1
Degree $21$
Signature $[21, 0]$
Discriminant $1.225\times 10^{41}$
Root discriminant \(90.48\)
Ramified primes $2,13,73,347443$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^6.C_{21}:C_3$ (as 21T86)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - x^19 + 114*x^18 - 219*x^17 - 472*x^16 + 1808*x^15 - 333*x^14 - 4832*x^13 + 5022*x^12 + 3881*x^11 - 8733*x^10 + 1621*x^9 + 5276*x^8 - 3399*x^7 - 644*x^6 + 1288*x^5 - 333*x^4 - 95*x^3 + 69*x^2 - 14*x + 1)
 
gp: K = bnfinit(y^21 - 7*y^20 - y^19 + 114*y^18 - 219*y^17 - 472*y^16 + 1808*y^15 - 333*y^14 - 4832*y^13 + 5022*y^12 + 3881*y^11 - 8733*y^10 + 1621*y^9 + 5276*y^8 - 3399*y^7 - 644*y^6 + 1288*y^5 - 333*y^4 - 95*y^3 + 69*y^2 - 14*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 7*x^20 - x^19 + 114*x^18 - 219*x^17 - 472*x^16 + 1808*x^15 - 333*x^14 - 4832*x^13 + 5022*x^12 + 3881*x^11 - 8733*x^10 + 1621*x^9 + 5276*x^8 - 3399*x^7 - 644*x^6 + 1288*x^5 - 333*x^4 - 95*x^3 + 69*x^2 - 14*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 7*x^20 - x^19 + 114*x^18 - 219*x^17 - 472*x^16 + 1808*x^15 - 333*x^14 - 4832*x^13 + 5022*x^12 + 3881*x^11 - 8733*x^10 + 1621*x^9 + 5276*x^8 - 3399*x^7 - 644*x^6 + 1288*x^5 - 333*x^4 - 95*x^3 + 69*x^2 - 14*x + 1)
 

\( x^{21} - 7 x^{20} - x^{19} + 114 x^{18} - 219 x^{17} - 472 x^{16} + 1808 x^{15} - 333 x^{14} - 4832 x^{13} + 5022 x^{12} + 3881 x^{11} - 8733 x^{10} + 1621 x^{9} + 5276 x^{8} - 3399 x^{7} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[21, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(122480818680759715483302268119489498578944\) \(\medspace = 2^{18}\cdot 13^{2}\cdot 73^{12}\cdot 347443^{2}\) 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:  \(90.48\)
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:  $2^{6/7}13^{2/3}73^{2/3}347443^{2/3}\approx 864554.2095127514$
Ramified primes:   \(2\), \(13\), \(73\), \(347443\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $3$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$ Copy content Toggle raw display

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

Monogenic:  Yes
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:  $20$
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:   $a-1$, $a$, $a^{19}-6a^{18}-7a^{17}+107a^{16}-112a^{15}-584a^{14}+1224a^{13}+891a^{12}-3941a^{11}+1081a^{10}+4962a^{9}-3771a^{8}-2150a^{7}+3126a^{6}-273a^{5}-917a^{4}+371a^{3}+38a^{2}-58a+13$, $1938a^{20}-12493a^{19}-8852a^{18}+216013a^{17}-304842a^{16}-1083196a^{15}+2903811a^{14}+960645a^{13}-8828668a^{12}+4847064a^{11}+10192466a^{10}-11277540a^{9}-3086153a^{8}+8503598a^{7}-1886941a^{6}-2281996a^{5}+1232378a^{4}+33456a^{3}-164471a^{2}+42956a-3548$, $19a^{20}-127a^{19}-57a^{18}+2135a^{17}-3500a^{16}-9842a^{15}+30977a^{14}+2202a^{13}-88309a^{12}+69367a^{11}+86593a^{10}-135890a^{9}-405a^{8}+91270a^{7}-41370a^{6}-17821a^{5}+18695a^{4}-2815a^{3}-1951a^{2}+892a-115$, $a^{20}-6a^{19}-7a^{18}+107a^{17}-112a^{16}-584a^{15}+1224a^{14}+891a^{13}-3941a^{12}+1081a^{11}+4962a^{10}-3771a^{9}-2150a^{8}+3126a^{7}-273a^{6}-917a^{5}+371a^{4}+38a^{3}-57a^{2}+11a-2$, $a^{19}-6a^{18}-7a^{17}+107a^{16}-112a^{15}-584a^{14}+1224a^{13}+891a^{12}-3941a^{11}+1081a^{10}+4962a^{9}-3771a^{8}-2150a^{7}+3126a^{6}-273a^{5}-917a^{4}+371a^{3}+38a^{2}-57a+11$, $19a^{20}-125a^{19}-69a^{18}+2121a^{17}-3286a^{16}-10066a^{15}+29809a^{14}+4650a^{13}-86527a^{12}+61485a^{11}+88755a^{10}-125966a^{9}-7947a^{8}+86970a^{7}-35118a^{6}-18367a^{5}+16861a^{4}-2073a^{3}-1875a^{2}+776a-91$, $1900a^{20}-12452a^{19}-7460a^{18}+213286a^{17}-320893a^{16}-1040293a^{15}+2971241a^{14}+694856a^{13}-8874121a^{12}+5579295a^{11}+9874944a^{10}-12189575a^{9}-2374176a^{8}+8976657a^{7}-2445773a^{6}-2324900a^{5}+1410391a^{4}-217a^{3}-181704a^{2}+49785a-4199$, $1938a^{20}-12741a^{19}-7364a^{18}+217807a^{17}-331668a^{16}-1056130a^{15}+3054209a^{14}+656334a^{13}-9085609a^{12}+5859640a^{11}+10019918a^{10}-12649671a^{9}-2252675a^{8}+9256972a^{7}-2636678a^{6}-2366258a^{5}+1483516a^{4}-14524a^{3}-188950a^{2}+53215a-4601$, $1746a^{20}-11199a^{19}-8302a^{18}+194148a^{17}-268672a^{16}-980941a^{15}+2581615a^{14}+927647a^{13}-7887533a^{12}+4154731a^{11}+9190653a^{10}-9866075a^{9}-2921425a^{8}+7492456a^{7}-1563545a^{6}-2031605a^{5}+1063401a^{4}+38396a^{3}-143586a^{2}+36729a-2980$, $1620a^{20}-10587a^{19}-6502a^{18}+181415a^{17}-270688a^{16}-886160a^{15}+2511878a^{14}+605081a^{13}-7493135a^{12}+4684233a^{11}+8292853a^{10}-10233537a^{9}-1909714a^{8}+7479910a^{7}-2134181a^{6}-1884418a^{5}+1206655a^{4}-27002a^{3}-152025a^{2}+45228a-4159$, $9733a^{20}-62718a^{19}-44610a^{18}+1084732a^{17}-1528286a^{16}-5443560a^{15}+14569559a^{14}+4859579a^{13}-44323574a^{12}+24233048a^{11}+51238363a^{10}-56502622a^{9}-15629011a^{8}+42651917a^{7}-9371985a^{6}-11472762a^{5}+6157202a^{4}+180623a^{3}-823765a^{2}+213694a-17521$, $1143a^{20}-7336a^{19}-5427a^{18}+127244a^{17}-176195a^{16}-643758a^{15}+1694172a^{14}+614300a^{13}-5187552a^{12}+2710196a^{11}+6082266a^{10}-6469828a^{9}-1998220a^{8}+4942805a^{7}-971349a^{6}-1361962a^{5}+683796a^{4}+35946a^{3}-93896a^{2}+22779a-1735$, $1667a^{20}-10818a^{19}-7188a^{18}+186361a^{17}-269958a^{16}-924484a^{15}+2542012a^{14}+741487a^{13}-7675635a^{12}+4456787a^{11}+8740760a^{10}-10101156a^{9}-2446240a^{8}+7550007a^{7}-1819755a^{6}-2003105a^{5}+1128116a^{4}+20320a^{3}-148657a^{2}+39294a-3250$, $3992a^{20}-26288a^{19}-14813a^{18}+448432a^{17}-688794a^{16}-2160803a^{15}+6311274a^{14}+1238318a^{13}-18668177a^{12}+12383724a^{11}+20277483a^{10}-26366756a^{9}-3998314a^{8}+19083289a^{7}-5896467a^{6}-4737817a^{5}+3191627a^{4}-97141a^{3}-398214a^{2}+118799a-10717$, $168a^{20}-1083a^{19}-759a^{18}+18678a^{17}-26500a^{16}-93013a^{15}+251060a^{14}+77947a^{13}-756513a^{12}+431818a^{11}+852176a^{10}-982145a^{9}-220863a^{8}+724443a^{7}-194990a^{6}-183048a^{5}+115501a^{4}-2616a^{3}-14733a^{2}+4393a-398$, $2368a^{20}-15323a^{19}-10428a^{18}+264148a^{17}-379177a^{16}-1313167a^{15}+3581437a^{14}+1077449a^{13}-10814445a^{12}+6220444a^{11}+12289836a^{10}-14148319a^{9}-3388535a^{8}+10549431a^{7}-2608740a^{6}-2770036a^{5}+1600785a^{4}+14098a^{3}-209342a^{2}+56782a-4794$, $1717a^{20}-11197a^{19}-7053a^{18}+192210a^{17}-284156a^{16}-943768a^{15}+2649375a^{14}+682022a^{13}-7937133a^{12}+4844171a^{11}+8876256a^{10}-10713004a^{9}-2206301a^{8}+7897719a^{7}-2124553a^{6}-2033546a^{5}+1239401a^{4}-7423a^{3}-159177a^{2}+44894a-3913$, $2575a^{20}-17165a^{19}-8314a^{18}+290814a^{17}-466773a^{16}-1371997a^{15}+4198417a^{14}+548237a^{13}-12269103a^{12}+8831008a^{11}+12973224a^{10}-18171319a^{9}-1931640a^{8}+12979845a^{7}-4402633a^{6}-3159960a^{5}+2264324a^{4}-91414a^{3}-278349a^{2}+83871a-7535$ 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:  \( 66161868019100 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{21}\cdot(2\pi)^{0}\cdot 66161868019100 \cdot 1}{2\cdot\sqrt{122480818680759715483302268119489498578944}}\cr\approx \mathstrut & 0.198231940175436 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - x^19 + 114*x^18 - 219*x^17 - 472*x^16 + 1808*x^15 - 333*x^14 - 4832*x^13 + 5022*x^12 + 3881*x^11 - 8733*x^10 + 1621*x^9 + 5276*x^8 - 3399*x^7 - 644*x^6 + 1288*x^5 - 333*x^4 - 95*x^3 + 69*x^2 - 14*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^21 - 7*x^20 - x^19 + 114*x^18 - 219*x^17 - 472*x^16 + 1808*x^15 - 333*x^14 - 4832*x^13 + 5022*x^12 + 3881*x^11 - 8733*x^10 + 1621*x^9 + 5276*x^8 - 3399*x^7 - 644*x^6 + 1288*x^5 - 333*x^4 - 95*x^3 + 69*x^2 - 14*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^21 - 7*x^20 - x^19 + 114*x^18 - 219*x^17 - 472*x^16 + 1808*x^15 - 333*x^14 - 4832*x^13 + 5022*x^12 + 3881*x^11 - 8733*x^10 + 1621*x^9 + 5276*x^8 - 3399*x^7 - 644*x^6 + 1288*x^5 - 333*x^4 - 95*x^3 + 69*x^2 - 14*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 7*x^20 - x^19 + 114*x^18 - 219*x^17 - 472*x^16 + 1808*x^15 - 333*x^14 - 4832*x^13 + 5022*x^12 + 3881*x^11 - 8733*x^10 + 1621*x^9 + 5276*x^8 - 3399*x^7 - 644*x^6 + 1288*x^5 - 333*x^4 - 95*x^3 + 69*x^2 - 14*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

$C_3^6.C_{21}:C_3$ (as 21T86):

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 solvable group of order 45927
The 183 conjugacy class representatives for $C_3^6.C_{21}:C_3$ are not computed
Character table for $C_3^6.C_{21}:C_3$ is not computed

Intermediate fields

7.7.1817487424.1

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]
 

Sibling fields

Degree 21 siblings: 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 $21$ ${\href{/padicField/5.9.0.1}{9} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ $21$ ${\href{/padicField/11.9.0.1}{9} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ R ${\href{/padicField/17.7.0.1}{7} }^{3}$ ${\href{/padicField/19.9.0.1}{9} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.9.0.1}{9} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ ${\href{/padicField/29.9.0.1}{9} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.9.0.1}{9} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ ${\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.3.0.1}{3} }^{4}$ ${\href{/padicField/41.9.0.1}{9} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }$ $21$ ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.9.0.1}{9} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }$

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
\(2\) Copy content Toggle raw display 2.21.18.1$x^{21} + 7 x^{19} + 7 x^{18} + 21 x^{17} + 42 x^{16} + 62 x^{15} + 111 x^{14} + 98 x^{13} - 189 x^{12} - 189 x^{11} + 259 x^{10} + 1496 x^{9} + 2586 x^{8} + 925 x^{7} + 798 x^{6} - 1092 x^{5} + 1029 x^{4} - 174 x^{3} - 53 x^{2} - 313 x + 131$$7$$3$$18$21T2$[\ ]_{7}^{3}$
\(13\) Copy content Toggle raw display 13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.9.0.1$x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$$1$$9$$0$$C_9$$[\ ]^{9}$
13.9.0.1$x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$$1$$9$$0$$C_9$$[\ ]^{9}$
\(73\) Copy content Toggle raw display 73.3.0.1$x^{3} + 2 x + 68$$1$$3$$0$$C_3$$[\ ]^{3}$
73.3.2.1$x^{3} + 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} + 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} + 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} + 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} + 73$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.1$x^{3} + 73$$3$$1$$2$$C_3$$[\ ]_{3}$
\(347443\) Copy content Toggle raw display Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$3$$1$$2$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$