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Properties

Label 22.2.162...408.1
Degree $22$
Signature $[2, 10]$
Discriminant $1.628\times 10^{30}$
Root discriminant \(23.62\)
Ramified primes $2,23,167$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^{10}.D_{22}$ (as 22T32)

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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^22 - 10*x^20 + 45*x^18 - 125*x^16 + 250*x^14 - 392*x^12 + 498*x^10 - 516*x^8 + 428*x^6 - 264*x^4 + 107*x^2 - 23)
 
Copy content gp:K = bnfinit(y^22 - 10*y^20 + 45*y^18 - 125*y^16 + 250*y^14 - 392*y^12 + 498*y^10 - 516*y^8 + 428*y^6 - 264*y^4 + 107*y^2 - 23, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 10*x^20 + 45*x^18 - 125*x^16 + 250*x^14 - 392*x^12 + 498*x^10 - 516*x^8 + 428*x^6 - 264*x^4 + 107*x^2 - 23);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^22 - 10*x^20 + 45*x^18 - 125*x^16 + 250*x^14 - 392*x^12 + 498*x^10 - 516*x^8 + 428*x^6 - 264*x^4 + 107*x^2 - 23)
 

\( x^{22} - 10 x^{20} + 45 x^{18} - 125 x^{16} + 250 x^{14} - 392 x^{12} + 498 x^{10} - 516 x^{8} + \cdots - 23 \) 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:  $22$
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:  $[2, 10]$
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:   \(1627617880014561481407655313408\) \(\medspace = 2^{22}\cdot 23\cdot 167^{10}\) 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:  \(23.62\)
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:   \(2\), \(23\), \(167\) 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:  \(\Q(\sqrt{23}) \)
$\Aut(K/\Q)$:   $C_2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{13}a^{20}-\frac{5}{13}a^{18}-\frac{6}{13}a^{16}+\frac{1}{13}a^{14}-\frac{5}{13}a^{12}-\frac{1}{13}a^{10}-\frac{1}{13}a^{8}-\frac{1}{13}a^{6}-\frac{6}{13}a^{4}+\frac{5}{13}a^{2}+\frac{2}{13}$, $\frac{1}{13}a^{21}-\frac{5}{13}a^{19}-\frac{6}{13}a^{17}+\frac{1}{13}a^{15}-\frac{5}{13}a^{13}-\frac{1}{13}a^{11}-\frac{1}{13}a^{9}-\frac{1}{13}a^{7}-\frac{6}{13}a^{5}+\frac{5}{13}a^{3}+\frac{2}{13}a$ 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:  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:  $11$
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:   $a^{2}-1$, $\frac{64}{13}a^{20}-\frac{541}{13}a^{18}+\frac{2047}{13}a^{16}-\frac{4863}{13}a^{14}+\frac{8572}{13}a^{12}-\frac{12011}{13}a^{10}+\frac{13560}{13}a^{8}-\frac{12362}{13}a^{6}+\frac{8547}{13}a^{4}-\frac{3853}{13}a^{2}+\frac{934}{13}$, $\frac{138}{13}a^{20}-\frac{1158}{13}a^{18}+\frac{4346}{13}a^{16}-\frac{10249}{13}a^{14}+\frac{17978}{13}a^{12}-\frac{25098}{13}a^{10}+\frac{28215}{13}a^{8}-\frac{25631}{13}a^{6}+\frac{17632}{13}a^{4}-\frac{7903}{13}a^{2}+\frac{1966}{13}$, $\frac{174}{13}a^{20}-\frac{1468}{13}a^{18}+\frac{5534}{13}a^{16}-\frac{13086}{13}a^{14}+\frac{22985}{13}a^{12}-\frac{32128}{13}a^{10}+\frac{36161}{13}a^{8}-\frac{32869}{13}a^{6}+\frac{22642}{13}a^{4}-\frac{10141}{13}a^{2}+\frac{2493}{13}$, $\frac{102}{13}a^{20}-\frac{861}{13}a^{18}+\frac{3249}{13}a^{16}-\frac{7698}{13}a^{14}+\frac{13556}{13}a^{12}-\frac{18991}{13}a^{10}+\frac{21426}{13}a^{8}-\frac{19537}{13}a^{6}+\frac{13519}{13}a^{4}-\frac{6120}{13}a^{2}+\frac{1530}{13}$, $a+1$, $\frac{40}{13}a^{20}-\frac{343}{13}a^{18}+\frac{1307}{13}a^{16}-\frac{3106}{13}a^{14}+\frac{5468}{13}a^{12}-\frac{7658}{13}a^{10}+\frac{8631}{13}a^{8}-\frac{7866}{13}a^{6}+\frac{5441}{13}a^{4}-\frac{2439}{13}a^{2}-a+\frac{613}{13}$, $\frac{174}{13}a^{21}-\frac{230}{13}a^{20}-\frac{1468}{13}a^{19}+\frac{1930}{13}a^{18}+\frac{5534}{13}a^{17}-\frac{7239}{13}a^{16}-\frac{13086}{13}a^{15}+\frac{17060}{13}a^{14}+\frac{22985}{13}a^{13}-\frac{29920}{13}a^{12}-\frac{32128}{13}a^{11}+\frac{41778}{13}a^{10}+\frac{36161}{13}a^{9}-\frac{46986}{13}a^{8}-\frac{32869}{13}a^{7}+\frac{42714}{13}a^{6}+\frac{22642}{13}a^{5}-\frac{29417}{13}a^{4}-\frac{10141}{13}a^{3}+\frac{13228}{13}a^{2}+\frac{2480}{13}a-\frac{3320}{13}$, $\frac{21}{13}a^{21}-\frac{34}{13}a^{20}-\frac{170}{13}a^{19}+\frac{287}{13}a^{18}+\frac{615}{13}a^{17}-\frac{1083}{13}a^{16}-\frac{1409}{13}a^{15}+\frac{2566}{13}a^{14}+\frac{2430}{13}a^{13}-\frac{4523}{13}a^{12}-\frac{3349}{13}a^{11}+\frac{6352}{13}a^{10}+\frac{3710}{13}a^{9}-\frac{7181}{13}a^{8}-\frac{3323}{13}a^{7}+\frac{6560}{13}a^{6}+\frac{2240}{13}a^{5}-\frac{4567}{13}a^{4}-\frac{974}{13}a^{3}+\frac{2092}{13}a^{2}+\frac{237}{13}a-\frac{536}{13}$, $\frac{125}{13}a^{21}+\frac{138}{13}a^{20}-\frac{1054}{13}a^{19}-\frac{1171}{13}a^{18}+\frac{3969}{13}a^{17}+\frac{4437}{13}a^{16}-\frac{9378}{13}a^{15}-\frac{10535}{13}a^{14}+\frac{16470}{13}a^{13}+\frac{18563}{13}a^{12}-\frac{23018}{13}a^{11}-\frac{26021}{13}a^{10}+\frac{25901}{13}a^{9}+\frac{29372}{13}a^{8}-\frac{23551}{13}a^{7}-\frac{26788}{13}a^{6}+\frac{16228}{13}a^{5}+\frac{18542}{13}a^{4}-\frac{7279}{13}a^{3}-\frac{8371}{13}a^{2}+\frac{1810}{13}a+\frac{2096}{13}$, $\frac{374}{13}a^{21}-\frac{428}{13}a^{20}-\frac{3157}{13}a^{19}+\frac{3609}{13}a^{18}+\frac{11913}{13}a^{17}-\frac{13604}{13}a^{16}-\frac{28213}{13}a^{15}+\frac{32189}{13}a^{14}+\frac{49636}{13}a^{13}-\frac{56594}{13}a^{12}-\frac{69482}{13}a^{11}+\frac{79169}{13}a^{10}+\frac{78341}{13}a^{9}-\frac{89194}{13}a^{8}-\frac{71367}{13}a^{7}+\frac{81197}{13}a^{6}+\frac{49301}{13}a^{5}-\frac{56049}{13}a^{4}-\frac{22219}{13}a^{3}+\frac{25251}{13}a^{2}+\frac{5519}{13}a-\frac{6290}{13}$ 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:  \( 1255829.15878 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 1255829.15878 \cdot 1}{2\cdot\sqrt{1627617880014561481407655313408}}\cr\approx \mathstrut & 0.188791751583 \end{aligned}\] (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^22 - 10*x^20 + 45*x^18 - 125*x^16 + 250*x^14 - 392*x^12 + 498*x^10 - 516*x^8 + 428*x^6 - 264*x^4 + 107*x^2 - 23) 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^22 - 10*x^20 + 45*x^18 - 125*x^16 + 250*x^14 - 392*x^12 + 498*x^10 - 516*x^8 + 428*x^6 - 264*x^4 + 107*x^2 - 23, 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^22 - 10*x^20 + 45*x^18 - 125*x^16 + 250*x^14 - 392*x^12 + 498*x^10 - 516*x^8 + 428*x^6 - 264*x^4 + 107*x^2 - 23); 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^22 - 10*x^20 + 45*x^18 - 125*x^16 + 250*x^14 - 392*x^12 + 498*x^10 - 516*x^8 + 428*x^6 - 264*x^4 + 107*x^2 - 23); 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

$C_2^{10}.D_{22}$ (as 22T32):

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 solvable group of order 45056
The 200 conjugacy class representatives for $C_2^{10}.D_{22}$
Character table for $C_2^{10}.D_{22}$

Intermediate fields

11.1.129891985607.1

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(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 22 siblings: data not computed
Degree 44 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 $22$ ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{7}$ ${\href{/padicField/7.11.0.1}{11} }^{2}$ ${\href{/padicField/11.11.0.1}{11} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{9}$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{7}$ ${\href{/padicField/19.11.0.1}{11} }^{2}$ R ${\href{/padicField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{7}$ ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ $22$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{7}$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.11.2.22a127.1$x^{22} + 2 x^{21} + 2 x^{19} + 2 x^{17} + 2 x^{15} + 2 x^{13} + 2 x^{12} + 2 x^{11} + 4 x^{10} + 4 x^{8} + 4 x^{6} + 3 x^{4} + 2 x^{2} + 3$$2$$11$$22$not computednot computed
\(23\) Copy content Toggle raw display 23.1.2.1a1.1$x^{2} + 23$$2$$1$$1$$C_2$$$[\ ]_{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.4.1.0a1.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
23.4.1.0a1.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
23.4.1.0a1.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$$[\ ]^{4}$$
\(167\) Copy content Toggle raw display 167.2.1.0a1.1$x^{2} + 166 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
167.1.2.1a1.1$x^{2} + 167$$2$$1$$1$$C_2$$$[\ ]_{2}$$
167.1.2.1a1.1$x^{2} + 167$$2$$1$$1$$C_2$$$[\ ]_{2}$$
167.1.2.1a1.1$x^{2} + 167$$2$$1$$1$$C_2$$$[\ ]_{2}$$
167.1.2.1a1.1$x^{2} + 167$$2$$1$$1$$C_2$$$[\ ]_{2}$$
167.1.2.1a1.1$x^{2} + 167$$2$$1$$1$$C_2$$$[\ ]_{2}$$
167.1.2.1a1.1$x^{2} + 167$$2$$1$$1$$C_2$$$[\ ]_{2}$$
167.2.2.2a1.2$x^{4} + 332 x^{3} + 27566 x^{2} + 1660 x + 192$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
167.2.2.2a1.2$x^{4} + 332 x^{3} + 27566 x^{2} + 1660 x + 192$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

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