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Properties

Label 23.5.250...508.1
Degree $23$
Signature $[5, 9]$
Discriminant $-2.503\times 10^{36}$
Root discriminant \(38.24\)
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 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 10*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 17*x^6 - 17*x^5 - 16*x^4 + 6*x^2 + x - 1)
 
gp: K = bnfinit(y^23 - 3*y^21 - 6*y^20 + y^19 + 17*y^18 + 18*y^17 - 10*y^16 - 43*y^15 - 25*y^14 + 33*y^13 + 60*y^12 + 10*y^11 - 54*y^10 - 47*y^9 + 12*y^8 + 44*y^7 + 17*y^6 - 17*y^5 - 16*y^4 + 6*y^2 + y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 10*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 17*x^6 - 17*x^5 - 16*x^4 + 6*x^2 + x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 10*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 17*x^6 - 17*x^5 - 16*x^4 + 6*x^2 + x - 1)
 

\( x^{23} - 3 x^{21} - 6 x^{20} + x^{19} + 17 x^{18} + 18 x^{17} - 10 x^{16} - 43 x^{15} - 25 x^{14} + \cdots - 1 \) 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:  $[5, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-2503137973364726455090108816728094508\) \(\medspace = -\,2^{2}\cdot 1307\cdot 1787\cdot 28687\cdot 512717\cdot 18216368638827725857\) 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:  \(38.24\)
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:  $2^{2/3}1307^{1/2}1787^{1/2}28687^{1/2}512717^{1/2}18216368638827725857^{1/2}\approx 1.2557380744871542e+18$
Ramified primes:   \(2\), \(1307\), \(1787\), \(28687\), \(512717\), \(18216368638827725857\) 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:  $\Q(\sqrt{-62578\!\cdots\!23627}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$ 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:  $13$
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:   $a^{22}-3a^{20}-6a^{19}+a^{18}+17a^{17}+18a^{16}-10a^{15}-43a^{14}-25a^{13}+33a^{12}+60a^{11}+10a^{10}-54a^{9}-47a^{8}+12a^{7}+44a^{6}+17a^{5}-17a^{4}-16a^{3}+6a+1$, $a^{21}-a^{20}-3a^{19}-2a^{18}+6a^{17}+13a^{16}-a^{15}-22a^{14}-20a^{13}+17a^{12}+36a^{11}+7a^{10}-33a^{9}-28a^{8}+14a^{7}+26a^{6}+4a^{5}-13a^{4}-8a^{3}+5a^{2}+3a-2$, $a^{21}-3a^{19}-6a^{18}+a^{17}+17a^{16}+18a^{15}-10a^{14}-43a^{13}-25a^{12}+33a^{11}+60a^{10}+10a^{9}-54a^{8}-47a^{7}+12a^{6}+44a^{5}+17a^{4}-16a^{3}-16a^{2}-a+5$, $a^{22}-3a^{20}-5a^{19}+a^{18}+14a^{17}+13a^{16}-9a^{15}-29a^{14}-12a^{13}+24a^{12}+31a^{11}-2a^{10}-30a^{9}-16a^{8}+10a^{7}+14a^{6}+a^{5}-7a^{4}-2a^{3}+a^{2}-a-1$, $4a^{22}+4a^{21}-12a^{20}-35a^{19}-20a^{18}+69a^{17}+134a^{16}+33a^{15}-195a^{14}-254a^{13}+22a^{12}+329a^{11}+255a^{10}-143a^{9}-344a^{8}-130a^{7}+170a^{6}+198a^{5}+11a^{4}-89a^{3}-48a^{2}+9a+13$, $7a^{22}+3a^{21}-19a^{20}-50a^{19}-16a^{18}+107a^{17}+171a^{16}+13a^{15}-280a^{14}-294a^{13}+78a^{12}+431a^{11}+263a^{10}-227a^{9}-409a^{8}-116a^{7}+228a^{6}+212a^{5}-7a^{4}-103a^{3}-49a^{2}+14a+12$, $9a^{22}+7a^{21}-25a^{20}-74a^{19}-36a^{18}+144a^{17}+269a^{16}+57a^{15}-399a^{14}-486a^{13}+61a^{12}+650a^{11}+461a^{10}-301a^{9}-649a^{8}-215a^{7}+334a^{6}+341a^{5}+2a^{4}-162a^{3}-70a^{2}+23a+14$, $13a^{22}+4a^{21}-35a^{20}-90a^{19}-22a^{18}+201a^{17}+303a^{16}+4a^{15}-526a^{14}-520a^{13}+175a^{12}+802a^{11}+459a^{10}-447a^{9}-754a^{8}-188a^{7}+442a^{6}+392a^{5}-24a^{4}-196a^{3}-90a^{2}+30a+22$, $a^{22}+a^{21}-3a^{20}-9a^{19}-5a^{18}+18a^{17}+35a^{16}+8a^{15}-53a^{14}-68a^{13}+8a^{12}+93a^{11}+70a^{10}-44a^{9}-100a^{8}-36a^{7}+54a^{6}+60a^{5}+2a^{4}-29a^{3}-15a^{2}+4a+5$, $16a^{22}-8a^{21}-41a^{20}-76a^{19}+46a^{18}+232a^{17}+176a^{16}-202a^{15}-540a^{14}-158a^{13}+493a^{12}+654a^{11}-80a^{10}-671a^{9}-400a^{8}+256a^{7}+465a^{6}+73a^{5}-205a^{4}-120a^{3}+22a^{2}+52a-11$, $14a^{22}+5a^{21}-40a^{20}-95a^{19}-22a^{18}+219a^{17}+317a^{16}-13a^{15}-554a^{14}-521a^{13}+218a^{12}+813a^{11}+423a^{10}-486a^{9}-730a^{8}-143a^{7}+433a^{6}+357a^{5}-41a^{4}-174a^{3}-70a^{2}+21a+18$, $5a^{22}+4a^{21}-13a^{20}-40a^{19}-23a^{18}+71a^{17}+143a^{16}+48a^{15}-187a^{14}-259a^{13}-11a^{12}+300a^{11}+264a^{10}-92a^{9}-315a^{8}-165a^{7}+120a^{6}+187a^{5}+39a^{4}-74a^{3}-53a^{2}+6a+12$, $80a^{22}+43a^{21}-217a^{20}-597a^{19}-240a^{18}+1232a^{17}+2103a^{16}+327a^{15}-3269a^{14}-3756a^{13}+629a^{12}+5146a^{11}+3560a^{10}-2418a^{9}-5065a^{8}-1752a^{7}+2588a^{6}+2750a^{5}+111a^{4}-1225a^{3}-655a^{2}+130a+150$ 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:  \( 3444212420.75 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{9}\cdot 3444212420.75 \cdot 1}{2\cdot\sqrt{2503137973364726455090108816728094508}}\cr\approx \mathstrut & 0.531601664558 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^23 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 10*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 17*x^6 - 17*x^5 - 16*x^4 + 6*x^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 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 10*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 17*x^6 - 17*x^5 - 16*x^4 + 6*x^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 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 10*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 17*x^6 - 17*x^5 - 16*x^4 + 6*x^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 - 3*x^21 - 6*x^20 + x^19 + 17*x^18 + 18*x^17 - 10*x^16 - 43*x^15 - 25*x^14 + 33*x^13 + 60*x^12 + 10*x^11 - 54*x^10 - 47*x^9 + 12*x^8 + 44*x^7 + 17*x^6 - 17*x^5 - 16*x^4 + 6*x^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 $S_{23}$
Character table for $S_{23}$

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 $16{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ $23$ $17{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.5.0.1}{5} }$ ${\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ $18{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ $23$ $19{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.9.0.1}{9} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $15{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ $19{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.13.0.1}{13} }{,}\,{\href{/padicField/47.10.0.1}{10} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x^{2} + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.12.0.1$x^{12} + x^{7} + x^{6} + x^{5} + x^{3} + x + 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(1307\) Copy content Toggle raw display $\Q_{1307}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $14$$1$$14$$0$$C_{14}$$[\ ]^{14}$
\(1787\) Copy content Toggle raw display $\Q_{1787}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(28687\) Copy content Toggle raw display $\Q_{28687}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{28687}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $12$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(512717\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $15$$1$$15$$0$$C_{15}$$[\ ]^{15}$
\(18216368638827725857\) Copy content Toggle raw display $\Q_{18216368638827725857}$$x$$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}$