Properties

Label 23.1.332...679.1
Degree $23$
Signature $[1, 11]$
Discriminant $-3.324\times 10^{38}$
Root discriminant \(47.30\)
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 - 9*x - 9)
 
gp: K = bnfinit(y^23 - 9*y - 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 9*x - 9);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 9*x - 9)
 

\( x^{23} - 9x - 9 \) 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:   \(-332434202154298684636783958613993659679\) \(\medspace = -\,3^{22}\cdot 103\cdot 409\cdot 136973\cdot 1835872831815206461\) 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:  \(47.30\)
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:  $3^{22/23}103^{1/2}409^{1/2}136973^{1/2}1835872831815206461^{1/2}\approx 294371594383291.06$
Ramified primes:   \(3\), \(103\), \(409\), \(136973\), \(1835872831815206461\) 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:  $\Q(\sqrt{-10593\!\cdots\!10231}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3}a^{12}$, $\frac{1}{3}a^{13}$, $\frac{1}{3}a^{14}$, $\frac{1}{3}a^{15}$, $\frac{1}{3}a^{16}$, $\frac{1}{3}a^{17}$, $\frac{1}{3}a^{18}$, $\frac{1}{3}a^{19}$, $\frac{1}{3}a^{20}$, $\frac{1}{3}a^{21}$, $\frac{1}{123}a^{22}+\frac{4}{41}a^{21}-\frac{20}{123}a^{20}+\frac{2}{41}a^{19}-\frac{10}{123}a^{18}+\frac{1}{41}a^{17}-\frac{5}{123}a^{16}-\frac{19}{123}a^{15}+\frac{6}{41}a^{14}+\frac{11}{123}a^{13}+\frac{3}{41}a^{12}-\frac{5}{41}a^{11}-\frac{19}{41}a^{10}+\frac{18}{41}a^{9}+\frac{11}{41}a^{8}+\frac{9}{41}a^{7}-\frac{15}{41}a^{6}-\frac{16}{41}a^{5}+\frac{13}{41}a^{4}-\frac{8}{41}a^{3}-\frac{14}{41}a^{2}-\frac{4}{41}a-\frac{10}{41}$ 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:   $\frac{1}{3}a^{22}-\frac{1}{3}a^{21}+\frac{1}{3}a^{20}-\frac{1}{3}a^{19}+\frac{1}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-2$, $\frac{1}{3}a^{22}-\frac{1}{3}a^{21}+\frac{1}{3}a^{20}-\frac{1}{3}a^{19}+\frac{1}{3}a^{18}-\frac{1}{3}a^{17}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}-4$, $\frac{31}{123}a^{22}-\frac{38}{123}a^{21}+\frac{12}{41}a^{20}-\frac{19}{123}a^{19}-\frac{23}{123}a^{18}+\frac{52}{123}a^{17}-\frac{73}{123}a^{16}+\frac{67}{123}a^{15}-\frac{16}{123}a^{14}-\frac{28}{123}a^{13}+\frac{74}{123}a^{12}-\frac{32}{41}a^{11}+\frac{26}{41}a^{10}-\frac{16}{41}a^{9}-\frac{28}{41}a^{8}+\frac{33}{41}a^{7}-\frac{55}{41}a^{6}+\frac{37}{41}a^{5}-\frac{7}{41}a^{4}-\frac{2}{41}a^{3}+\frac{58}{41}a^{2}-\frac{42}{41}a-\frac{23}{41}$, $\frac{16}{123}a^{22}-\frac{13}{123}a^{21}+\frac{49}{123}a^{20}-\frac{9}{41}a^{19}+\frac{15}{41}a^{18}+\frac{7}{123}a^{17}+\frac{43}{123}a^{16}-\frac{17}{123}a^{15}+\frac{14}{41}a^{14}+\frac{4}{41}a^{13}-\frac{20}{123}a^{12}+\frac{2}{41}a^{11}-\frac{17}{41}a^{10}+\frac{1}{41}a^{9}-\frac{29}{41}a^{8}-\frac{20}{41}a^{7}-\frac{35}{41}a^{6}-\frac{10}{41}a^{5}-\frac{38}{41}a^{4}-\frac{46}{41}a^{3}+\frac{22}{41}a^{2}-\frac{23}{41}a-\frac{37}{41}$, $\frac{6}{41}a^{22}-\frac{10}{41}a^{21}+\frac{3}{41}a^{20}+\frac{26}{123}a^{19}-\frac{19}{41}a^{18}+\frac{18}{41}a^{17}-\frac{8}{123}a^{16}-\frac{55}{123}a^{15}+\frac{26}{41}a^{14}-\frac{16}{41}a^{13}-\frac{43}{123}a^{12}+\frac{33}{41}a^{11}-\frac{14}{41}a^{10}-\frac{4}{41}a^{9}+\frac{34}{41}a^{8}-\frac{43}{41}a^{7}+\frac{17}{41}a^{6}-\frac{1}{41}a^{5}-\frac{53}{41}a^{4}+\frac{61}{41}a^{3}-\frac{6}{41}a^{2}-\frac{31}{41}a+\frac{25}{41}$, $\frac{32}{41}a^{22}-\frac{119}{123}a^{21}+\frac{89}{123}a^{20}-\frac{13}{41}a^{19}+\frac{8}{41}a^{18}+\frac{1}{123}a^{17}-\frac{70}{123}a^{16}+\frac{48}{41}a^{15}-\frac{39}{41}a^{14}+\frac{113}{123}a^{13}-\frac{202}{123}a^{12}+\frac{53}{41}a^{11}-\frac{20}{41}a^{10}+\frac{6}{41}a^{9}-\frac{10}{41}a^{8}-\frac{38}{41}a^{7}+\frac{77}{41}a^{6}-\frac{60}{41}a^{5}+\frac{59}{41}a^{4}-\frac{112}{41}a^{3}+\frac{91}{41}a^{2}-\frac{15}{41}a-\frac{263}{41}$, $\frac{8}{41}a^{22}-\frac{40}{123}a^{21}+\frac{4}{41}a^{20}+\frac{7}{41}a^{19}-\frac{35}{123}a^{18}-\frac{10}{123}a^{17}+\frac{44}{123}a^{16}-\frac{46}{123}a^{15}+\frac{22}{123}a^{14}+\frac{6}{41}a^{13}+\frac{11}{123}a^{12}+\frac{3}{41}a^{11}-\frac{5}{41}a^{10}+\frac{22}{41}a^{9}-\frac{23}{41}a^{8}-\frac{30}{41}a^{7}+\frac{50}{41}a^{6}-\frac{56}{41}a^{5}-\frac{16}{41}a^{4}+\frac{95}{41}a^{3}-\frac{49}{41}a^{2}-\frac{14}{41}a+\frac{47}{41}$, $\frac{4}{123}a^{22}-\frac{34}{123}a^{21}+\frac{43}{123}a^{20}+\frac{8}{41}a^{19}-\frac{40}{123}a^{18}-\frac{29}{123}a^{17}+\frac{7}{41}a^{16}+\frac{2}{41}a^{15}+\frac{31}{123}a^{14}+\frac{44}{123}a^{13}-\frac{128}{123}a^{12}-\frac{20}{41}a^{11}+\frac{47}{41}a^{10}+\frac{31}{41}a^{9}-\frac{38}{41}a^{8}-\frac{5}{41}a^{7}-\frac{19}{41}a^{6}-\frac{23}{41}a^{5}+\frac{52}{41}a^{4}+\frac{50}{41}a^{3}-\frac{56}{41}a^{2}-\frac{16}{41}a+\frac{1}{41}$, $\frac{37}{123}a^{22}-\frac{16}{41}a^{21}+\frac{13}{41}a^{20}-\frac{8}{41}a^{19}-\frac{1}{123}a^{18}-\frac{53}{123}a^{17}+\frac{61}{123}a^{16}-\frac{47}{123}a^{15}+\frac{10}{123}a^{14}-\frac{1}{41}a^{13}+\frac{5}{123}a^{12}-\frac{21}{41}a^{11}+\frac{35}{41}a^{10}+\frac{10}{41}a^{9}-\frac{3}{41}a^{8}+\frac{5}{41}a^{7}+\frac{19}{41}a^{6}-\frac{18}{41}a^{5}+\frac{30}{41}a^{4}+\frac{32}{41}a^{3}-\frac{26}{41}a^{2}-\frac{25}{41}a-\frac{83}{41}$, $\frac{40}{123}a^{22}-\frac{53}{123}a^{21}-\frac{7}{41}a^{20}+\frac{35}{123}a^{19}-\frac{113}{123}a^{18}+\frac{40}{41}a^{17}-\frac{200}{123}a^{16}+\frac{142}{123}a^{15}-\frac{223}{123}a^{14}+\frac{51}{41}a^{13}-\frac{173}{123}a^{12}+\frac{5}{41}a^{11}-\frac{22}{41}a^{10}-\frac{59}{41}a^{9}+\frac{30}{41}a^{8}-\frac{91}{41}a^{7}+\frac{56}{41}a^{6}-\frac{148}{41}a^{5}+\frac{28}{41}a^{4}-\frac{115}{41}a^{3}-\frac{27}{41}a^{2}-\frac{37}{41}a-\frac{236}{41}$, $\frac{94}{123}a^{22}-\frac{143}{123}a^{21}+\frac{211}{123}a^{20}-\frac{215}{123}a^{19}+\frac{208}{123}a^{18}-\frac{70}{41}a^{17}+\frac{145}{123}a^{16}-\frac{35}{41}a^{15}-\frac{10}{41}a^{14}+\frac{44}{41}a^{13}-\frac{220}{123}a^{12}+\frac{104}{41}a^{11}-\frac{105}{41}a^{10}+\frac{134}{41}a^{9}-\frac{114}{41}a^{8}+\frac{108}{41}a^{7}-\frac{57}{41}a^{6}-\frac{28}{41}a^{5}+\frac{33}{41}a^{4}-\frac{137}{41}a^{3}+\frac{119}{41}a^{2}-\frac{171}{41}a-\frac{38}{41}$ 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:  \( 12917754140.4 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 12917754140.4 \cdot 1}{2\cdot\sqrt{332434202154298684636783958613993659679}}\cr\approx \mathstrut & 0.426886790957 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^23 - 9*x - 9)
 
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 - 9*x - 9, 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 - 9*x - 9);
 
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 - 9*x - 9);
 
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 ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ R $16{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ $20{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.7.0.1}{7} }^{2}{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.11.0.1}{11} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $19{,}\,{\href{/padicField/29.4.0.1}{4} }$ $18{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.7.0.1}{7} }^{2}{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ $18{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ $17{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ $19{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(3\) Copy content Toggle raw display 3.23.22.1$x^{23} + 3$$23$$1$$22$$C_{23}:C_{11}$$[\ ]_{23}^{11}$
\(103\) Copy content Toggle raw display 103.2.1.2$x^{2} + 103$$2$$1$$1$$C_2$$[\ ]_{2}$
103.4.0.1$x^{4} + 2 x^{2} + 88 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
103.17.0.1$x^{17} + 102 x^{2} + 8 x + 98$$1$$17$$0$$C_{17}$$[\ ]^{17}$
\(409\) Copy content Toggle raw display $\Q_{409}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $7$$1$$7$$0$$C_7$$[\ ]^{7}$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$
\(136973\) Copy content Toggle raw display $\Q_{136973}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{136973}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{136973}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{136973}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
\(1835872831815206461\) Copy content Toggle raw display $\Q_{1835872831815206461}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $15$$1$$15$$0$$C_{15}$$[\ ]^{15}$