Properties

Label 23.1.459...108.1
Degree $23$
Signature $[1, 11]$
Discriminant $-4.592\times 10^{35}$
Root discriminant \(35.52\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{23}$ (as 23T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 2*x^22 - x^21 + 5*x^20 - x^19 - 4*x^18 + 6*x^15 + 6*x^14 - 17*x^13 - 10*x^12 + 31*x^11 + 5*x^10 - 32*x^9 + 25*x^7 - 17*x^5 + 2*x^4 + 9*x^3 - 3*x^2 - 2*x + 1)
 
gp: K = bnfinit(y^23 - 2*y^22 - y^21 + 5*y^20 - y^19 - 4*y^18 + 6*y^15 + 6*y^14 - 17*y^13 - 10*y^12 + 31*y^11 + 5*y^10 - 32*y^9 + 25*y^7 - 17*y^5 + 2*y^4 + 9*y^3 - 3*y^2 - 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 2*x^22 - x^21 + 5*x^20 - x^19 - 4*x^18 + 6*x^15 + 6*x^14 - 17*x^13 - 10*x^12 + 31*x^11 + 5*x^10 - 32*x^9 + 25*x^7 - 17*x^5 + 2*x^4 + 9*x^3 - 3*x^2 - 2*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 2*x^22 - x^21 + 5*x^20 - x^19 - 4*x^18 + 6*x^15 + 6*x^14 - 17*x^13 - 10*x^12 + 31*x^11 + 5*x^10 - 32*x^9 + 25*x^7 - 17*x^5 + 2*x^4 + 9*x^3 - 3*x^2 - 2*x + 1)
 

\( x^{23} - 2 x^{22} - x^{21} + 5 x^{20} - x^{19} - 4 x^{18} + 6 x^{15} + 6 x^{14} - 17 x^{13} - 10 x^{12} + 31 x^{11} + 5 x^{10} - 32 x^{9} + 25 x^{7} - 17 x^{5} + 2 x^{4} + 9 x^{3} - 3 x^{2} - 2 x + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $23$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-459177995857290463522143518056811108\) \(\medspace = -\,2^{2}\cdot 1021\cdot 425123\cdot 264472629367076822658099919\) 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:  \(35.52\)
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:  not computed
Ramified primes:   \(2\), \(1021\), \(425123\), \(264472629367076822658099919\) 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:  $\Q(\sqrt{-11479\!\cdots\!02777}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{11}a^{22}+\frac{4}{11}a^{21}+\frac{1}{11}a^{20}-\frac{1}{11}a^{18}+\frac{1}{11}a^{17}-\frac{5}{11}a^{16}+\frac{3}{11}a^{15}+\frac{2}{11}a^{14}-\frac{4}{11}a^{13}+\frac{3}{11}a^{12}-\frac{3}{11}a^{11}+\frac{2}{11}a^{10}-\frac{5}{11}a^{9}+\frac{4}{11}a^{8}+\frac{2}{11}a^{7}+\frac{4}{11}a^{6}+\frac{2}{11}a^{5}-\frac{5}{11}a^{4}+\frac{5}{11}a^{3}-\frac{5}{11}a^{2}-\frac{2}{11}$ 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:  $11$
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:   $a$, $a^{22}-3a^{21}+a^{20}+5a^{19}-4a^{18}-2a^{17}+8a^{14}+2a^{13}-21a^{12}+a^{11}+33a^{10}-13a^{9}-25a^{8}+11a^{7}+17a^{6}-8a^{5}-11a^{4}+8a^{3}+4a^{2}-4a+1$, $\frac{90}{11}a^{22}-\frac{102}{11}a^{21}-\frac{185}{11}a^{20}+26a^{19}+\frac{185}{11}a^{18}-\frac{196}{11}a^{17}-\frac{219}{11}a^{16}-\frac{203}{11}a^{15}+\frac{400}{11}a^{14}+\frac{938}{11}a^{13}-\frac{709}{11}a^{12}-\frac{1634}{11}a^{11}+\frac{1291}{11}a^{10}+\frac{1761}{11}a^{9}-\frac{1224}{11}a^{8}-\frac{1316}{11}a^{7}+\frac{976}{11}a^{6}+\frac{1049}{11}a^{5}-\frac{527}{11}a^{4}-\frac{430}{11}a^{3}+\frac{364}{11}a^{2}+11a-\frac{59}{11}$, $6a^{22}-16a^{21}+5a^{20}+26a^{19}-24a^{18}-6a^{17}+4a^{16}-4a^{15}+38a^{14}+10a^{13}-106a^{12}+14a^{11}+171a^{10}-90a^{9}-122a^{8}+86a^{7}+83a^{6}-58a^{5}-58a^{4}+53a^{3}+14a^{2}-27a+8$, $\frac{18}{11}a^{22}-\frac{27}{11}a^{21}-\frac{48}{11}a^{20}+9a^{19}+\frac{37}{11}a^{18}-\frac{103}{11}a^{17}-\frac{46}{11}a^{16}-\frac{12}{11}a^{15}+\frac{146}{11}a^{14}+\frac{203}{11}a^{13}-\frac{298}{11}a^{12}-\frac{439}{11}a^{11}+\frac{531}{11}a^{10}+\frac{504}{11}a^{9}-\frac{599}{11}a^{8}-\frac{382}{11}a^{7}+\frac{402}{11}a^{6}+\frac{333}{11}a^{5}-\frac{266}{11}a^{4}-\frac{174}{11}a^{3}+\frac{130}{11}a^{2}+2a-\frac{14}{11}$, $\frac{34}{11}a^{22}-\frac{40}{11}a^{21}-\frac{65}{11}a^{20}+10a^{19}+\frac{54}{11}a^{18}-\frac{76}{11}a^{17}-\frac{60}{11}a^{16}-\frac{63}{11}a^{15}+\frac{134}{11}a^{14}+\frac{315}{11}a^{13}-\frac{283}{11}a^{12}-\frac{542}{11}a^{11}+\frac{552}{11}a^{10}+\frac{556}{11}a^{9}-\frac{557}{11}a^{8}-\frac{394}{11}a^{7}+\frac{466}{11}a^{6}+\frac{321}{11}a^{5}-\frac{280}{11}a^{4}-\frac{127}{11}a^{3}+\frac{171}{11}a^{2}+3a-\frac{46}{11}$, $\frac{17}{11}a^{22}-\frac{20}{11}a^{21}-\frac{38}{11}a^{20}+4a^{19}+\frac{60}{11}a^{18}-\frac{27}{11}a^{17}-\frac{74}{11}a^{16}-\frac{59}{11}a^{15}+\frac{67}{11}a^{14}+\frac{229}{11}a^{13}-\frac{81}{11}a^{12}-\frac{370}{11}a^{11}+\frac{122}{11}a^{10}+\frac{399}{11}a^{9}-\frac{108}{11}a^{8}-\frac{340}{11}a^{7}+\frac{112}{11}a^{6}+\frac{210}{11}a^{5}-\frac{41}{11}a^{4}-\frac{124}{11}a^{3}+\frac{36}{11}a^{2}+4a-\frac{23}{11}$, $\frac{73}{11}a^{22}-\frac{93}{11}a^{21}-\frac{180}{11}a^{20}+27a^{19}+\frac{202}{11}a^{18}-\frac{290}{11}a^{17}-\frac{255}{11}a^{16}-\frac{89}{11}a^{15}+\frac{476}{11}a^{14}+\frac{852}{11}a^{13}-\frac{859}{11}a^{12}-\frac{1726}{11}a^{11}+\frac{1466}{11}a^{10}+\frac{2077}{11}a^{9}-\frac{1556}{11}a^{8}-\frac{1768}{11}a^{7}+\frac{1161}{11}a^{6}+\frac{1367}{11}a^{5}-\frac{662}{11}a^{4}-\frac{735}{11}a^{3}+\frac{350}{11}a^{2}+18a-\frac{91}{11}$, $\frac{80}{11}a^{22}-\frac{109}{11}a^{21}-\frac{129}{11}a^{20}+24a^{19}+\frac{118}{11}a^{18}-\frac{206}{11}a^{17}-\frac{169}{11}a^{16}-\frac{90}{11}a^{15}+\frac{369}{11}a^{14}+\frac{714}{11}a^{13}-\frac{805}{11}a^{12}-\frac{1252}{11}a^{11}+\frac{1425}{11}a^{10}+\frac{1327}{11}a^{9}-\frac{1407}{11}a^{8}-\frac{1061}{11}a^{7}+\frac{1255}{11}a^{6}+\frac{776}{11}a^{5}-\frac{708}{11}a^{4}-\frac{381}{11}a^{3}+\frac{447}{11}a^{2}+8a-\frac{127}{11}$, $4a^{22}-4a^{21}-12a^{20}+13a^{19}+18a^{18}-13a^{17}-20a^{16}-11a^{15}+24a^{14}+59a^{13}-31a^{12}-115a^{11}+46a^{10}+145a^{9}-47a^{8}-127a^{7}+29a^{6}+89a^{5}-11a^{4}-52a^{3}+6a^{2}+13a-2$, $\frac{59}{11}a^{22}-\frac{94}{11}a^{21}-\frac{106}{11}a^{20}+24a^{19}+\frac{73}{11}a^{18}-\frac{260}{11}a^{17}-\frac{108}{11}a^{16}+\frac{1}{11}a^{15}+\frac{349}{11}a^{14}+\frac{512}{11}a^{13}-\frac{835}{11}a^{12}-\frac{1035}{11}a^{11}+\frac{1548}{11}a^{10}+\frac{1102}{11}a^{9}-\frac{1733}{11}a^{8}-\frac{817}{11}a^{7}+\frac{1446}{11}a^{6}+\frac{580}{11}a^{5}-\frac{955}{11}a^{4}-\frac{255}{11}a^{3}+\frac{486}{11}a^{2}+a-\frac{140}{11}$ 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:  \( 583250036.6 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 583250036.6 \cdot 1}{2\cdot\sqrt{459177995857290463522143518056811108}}\cr\approx \mathstrut & 0.5186122759 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^23 - 2*x^22 - x^21 + 5*x^20 - x^19 - 4*x^18 + 6*x^15 + 6*x^14 - 17*x^13 - 10*x^12 + 31*x^11 + 5*x^10 - 32*x^9 + 25*x^7 - 17*x^5 + 2*x^4 + 9*x^3 - 3*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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^23 - 2*x^22 - x^21 + 5*x^20 - x^19 - 4*x^18 + 6*x^15 + 6*x^14 - 17*x^13 - 10*x^12 + 31*x^11 + 5*x^10 - 32*x^9 + 25*x^7 - 17*x^5 + 2*x^4 + 9*x^3 - 3*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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^23 - 2*x^22 - x^21 + 5*x^20 - x^19 - 4*x^18 + 6*x^15 + 6*x^14 - 17*x^13 - 10*x^12 + 31*x^11 + 5*x^10 - 32*x^9 + 25*x^7 - 17*x^5 + 2*x^4 + 9*x^3 - 3*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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 2*x^22 - x^21 + 5*x^20 - x^19 - 4*x^18 + 6*x^15 + 6*x^14 - 17*x^13 - 10*x^12 + 31*x^11 + 5*x^10 - 32*x^9 + 25*x^7 - 17*x^5 + 2*x^4 + 9*x^3 - 3*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):

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 non-solvable group of order 25852016738884976640000
The 1255 conjugacy class representatives for $S_{23}$ are not computed
Character table for $S_{23}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 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 $17{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ $21{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.14.0.1}{14} }{,}\,{\href{/padicField/7.9.0.1}{9} }$ $17{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.4.0.1}{4} }$ ${\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ $20{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ $18{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ $15{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.21.0.1$x^{21} + x^{6} + x^{5} + x^{2} + 1$$1$$21$$0$$C_{21}$$[\ ]^{21}$
\(1021\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $18$$1$$18$$0$$C_{18}$$[\ ]^{18}$
\(425123\) Copy content Toggle raw display $\Q_{425123}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$
\(264\!\cdots\!919\) Copy content Toggle raw display $\Q_{26\!\cdots\!19}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $17$$1$$17$$0$$C_{17}$$[\ ]^{17}$