Properties

Label 23.9.300...043.1
Degree $23$
Signature $[9, 7]$
Discriminant $-3.004\times 10^{40}$
Root discriminant \(57.53\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{23}$ (as 23T7)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 2*x^22 - 4*x^21 + 6*x^20 + 10*x^19 - 7*x^18 - 14*x^17 + 11*x^16 + 12*x^15 - 31*x^14 - 36*x^13 + 41*x^12 + 89*x^11 + 5*x^10 - 105*x^9 - 65*x^8 + 56*x^7 + 68*x^6 - 7*x^5 - 34*x^4 - 4*x^3 + 9*x^2 + x - 1)
 
gp: K = bnfinit(y^23 - 2*y^22 - 4*y^21 + 6*y^20 + 10*y^19 - 7*y^18 - 14*y^17 + 11*y^16 + 12*y^15 - 31*y^14 - 36*y^13 + 41*y^12 + 89*y^11 + 5*y^10 - 105*y^9 - 65*y^8 + 56*y^7 + 68*y^6 - 7*y^5 - 34*y^4 - 4*y^3 + 9*y^2 + y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 2*x^22 - 4*x^21 + 6*x^20 + 10*x^19 - 7*x^18 - 14*x^17 + 11*x^16 + 12*x^15 - 31*x^14 - 36*x^13 + 41*x^12 + 89*x^11 + 5*x^10 - 105*x^9 - 65*x^8 + 56*x^7 + 68*x^6 - 7*x^5 - 34*x^4 - 4*x^3 + 9*x^2 + x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 2*x^22 - 4*x^21 + 6*x^20 + 10*x^19 - 7*x^18 - 14*x^17 + 11*x^16 + 12*x^15 - 31*x^14 - 36*x^13 + 41*x^12 + 89*x^11 + 5*x^10 - 105*x^9 - 65*x^8 + 56*x^7 + 68*x^6 - 7*x^5 - 34*x^4 - 4*x^3 + 9*x^2 + x - 1)
 

\( x^{23} - 2 x^{22} - 4 x^{21} + 6 x^{20} + 10 x^{19} - 7 x^{18} - 14 x^{17} + 11 x^{16} + 12 x^{15} - 31 x^{14} - 36 x^{13} + 41 x^{12} + 89 x^{11} + 5 x^{10} - 105 x^{9} - 65 x^{8} + 56 x^{7} + 68 x^{6} + \cdots - 1 \) 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:  $[9, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-30040962001154759467875853831400403771043\) \(\medspace = -\,508621\cdot 158113511\cdot 1092887040821\cdot 341802543194293\) 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:  \(57.53\)
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:  $508621^{1/2}158113511^{1/2}1092887040821^{1/2}341802543194293^{1/2}\approx 1.7332328753273392e+20$
Ramified primes:   \(508621\), \(158113511\), \(1092887040821\), \(341802543194293\) 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:  $\Q(\sqrt{-30040\!\cdots\!71043}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:  $15$
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:   $a$, $a^{21}-2a^{20}-3a^{19}+4a^{18}+7a^{17}-3a^{16}-7a^{15}+8a^{14}+5a^{13}-23a^{12}-31a^{11}+18a^{10}+58a^{9}+23a^{8}-47a^{7}-42a^{6}+9a^{5}+26a^{4}+2a^{3}-8a^{2}-2a+1$, $a^{22}-2a^{21}-4a^{20}+6a^{19}+10a^{18}-7a^{17}-14a^{16}+11a^{15}+12a^{14}-31a^{13}-36a^{12}+41a^{11}+89a^{10}+5a^{9}-105a^{8}-65a^{7}+56a^{6}+68a^{5}-7a^{4}-34a^{3}-4a^{2}+9a$, $a^{22}-a^{21}-6a^{20}+2a^{19}+16a^{18}+3a^{17}-21a^{16}-3a^{15}+23a^{14}-19a^{13}-67a^{12}+5a^{11}+130a^{10}+94a^{9}-100a^{8}-170a^{7}-9a^{6}+124a^{5}+61a^{4}-41a^{3}-38a^{2}+4a+9$, $3a^{22}+a^{21}-26a^{20}-9a^{19}+70a^{18}+45a^{17}-85a^{16}-55a^{15}+106a^{14}-23a^{13}-314a^{12}-117a^{11}+523a^{10}+602a^{9}-239a^{8}-841a^{7}-282a^{6}+491a^{5}+390a^{4}-95a^{3}-182a^{2}-8a+33$, $33a^{22}-51a^{21}-188a^{20}+178a^{19}+523a^{18}-150a^{17}-793a^{16}+134a^{15}+762a^{14}-950a^{13}-1833a^{12}+1366a^{11}+4619a^{10}+1437a^{9}-5120a^{8}-5174a^{7}+1564a^{6}+4735a^{5}+1369a^{4}-1701a^{3}-1066a^{2}+199a+184$, $79a^{22}-125a^{21}-347a^{20}+288a^{19}+833a^{18}-103a^{17}-966a^{16}+378a^{15}+888a^{14}-1901a^{13}-3472a^{12}+1227a^{11}+6816a^{10}+3799a^{9}-5123a^{8}-6801a^{7}+128a^{6}+4173a^{5}+1615a^{4}-1122a^{3}-678a^{2}+124a+73$, $8a^{22}-10a^{21}-43a^{20}+23a^{19}+110a^{18}+6a^{17}-138a^{16}+7a^{15}+141a^{14}-179a^{13}-451a^{12}+93a^{11}+891a^{10}+579a^{9}-680a^{8}-1056a^{7}-42a^{6}+710a^{5}+342a^{4}-191a^{3}-173a^{2}+13a+27$, $156a^{22}-227a^{21}-747a^{20}+529a^{19}+1844a^{18}-90a^{17}-2225a^{16}+515a^{15}+2150a^{14}-3672a^{13}-7610a^{12}+2244a^{11}+15057a^{10}+8934a^{9}-11469a^{8}-16268a^{7}-68a^{6}+10501a^{5}+4527a^{4}-2856a^{3}-2139a^{2}+266a+298$, $70a^{22}-106a^{21}-323a^{20}+246a^{19}+790a^{18}-63a^{17}-941a^{16}+272a^{15}+887a^{14}-1660a^{13}-3265a^{12}+1055a^{11}+6472a^{10}+3739a^{9}-4920a^{8}-6808a^{7}+9a^{6}+4308a^{5}+1821a^{4}-1147a^{3}-819a^{2}+103a+106$, $191a^{22}-271a^{21}-911a^{20}+597a^{19}+2218a^{18}-a^{17}-2585a^{16}+566a^{15}+2518a^{14}-4381a^{13}-9341a^{12}+2136a^{11}+17869a^{10}+11565a^{9}-12557a^{8}-19410a^{7}-1229a^{6}+11637a^{5}+5550a^{4}-2867a^{3}-2350a^{2}+236a+300$, $64a^{22}-125a^{21}-241a^{20}+330a^{19}+582a^{18}-309a^{17}-734a^{16}+560a^{15}+571a^{14}-1762a^{13}-2225a^{12}+1940a^{11}+5115a^{10}+1225a^{9}-5078a^{8}-4102a^{7}+1778a^{6}+3251a^{5}+328a^{4}-1198a^{3}-275a^{2}+193a+18$, $58a^{22}-38a^{21}-298a^{20}-72a^{19}+609a^{18}+543a^{17}-340a^{16}-214a^{15}+529a^{14}-767a^{13}-3416a^{12}-2242a^{11}+3692a^{10}+6959a^{9}+1969a^{8}-4989a^{7}-5219a^{6}-449a^{5}+2217a^{4}+1247a^{3}-43a^{2}-239a-64$, $122a^{22}-165a^{21}-621a^{20}+392a^{19}+1543a^{18}-8a^{17}-1888a^{16}+297a^{15}+1881a^{14}-2854a^{13}-6348a^{12}+1607a^{11}+12534a^{10}+7740a^{9}-9481a^{8}-14024a^{7}-265a^{6}+9125a^{5}+4100a^{4}-2489a^{3}-1976a^{2}+230a+281$, $2a^{22}+32a^{21}-37a^{20}-177a^{19}+16a^{18}+400a^{17}+225a^{16}-306a^{15}-67a^{14}+355a^{13}-642a^{12}-2026a^{11}-717a^{10}+2804a^{9}+3669a^{8}-95a^{7}-3478a^{6}-2281a^{5}+664a^{4}+1357a^{3}+319a^{2}-210a-86$ 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:  \( 394956948058 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{7}\cdot 394956948058 \cdot 1}{2\cdot\sqrt{30040962001154759467875853831400403771043}}\cr\approx \mathstrut & 0.225523542178391 \end{aligned}\] (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 - 4*x^21 + 6*x^20 + 10*x^19 - 7*x^18 - 14*x^17 + 11*x^16 + 12*x^15 - 31*x^14 - 36*x^13 + 41*x^12 + 89*x^11 + 5*x^10 - 105*x^9 - 65*x^8 + 56*x^7 + 68*x^6 - 7*x^5 - 34*x^4 - 4*x^3 + 9*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 - 2*x^22 - 4*x^21 + 6*x^20 + 10*x^19 - 7*x^18 - 14*x^17 + 11*x^16 + 12*x^15 - 31*x^14 - 36*x^13 + 41*x^12 + 89*x^11 + 5*x^10 - 105*x^9 - 65*x^8 + 56*x^7 + 68*x^6 - 7*x^5 - 34*x^4 - 4*x^3 + 9*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 - 2*x^22 - 4*x^21 + 6*x^20 + 10*x^19 - 7*x^18 - 14*x^17 + 11*x^16 + 12*x^15 - 31*x^14 - 36*x^13 + 41*x^12 + 89*x^11 + 5*x^10 - 105*x^9 - 65*x^8 + 56*x^7 + 68*x^6 - 7*x^5 - 34*x^4 - 4*x^3 + 9*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 - 2*x^22 - 4*x^21 + 6*x^20 + 10*x^19 - 7*x^18 - 14*x^17 + 11*x^16 + 12*x^15 - 31*x^14 - 36*x^13 + 41*x^12 + 89*x^11 + 5*x^10 - 105*x^9 - 65*x^8 + 56*x^7 + 68*x^6 - 7*x^5 - 34*x^4 - 4*x^3 + 9*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 $S_{23}$ 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 ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.10.0.1}{10} }$ ${\href{/padicField/3.13.0.1}{13} }{,}\,{\href{/padicField/3.10.0.1}{10} }$ $21{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.9.0.1}{9} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ $18{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ $16{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ $16{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ $21{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ $23$ $16{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ $22{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(508621\) Copy content Toggle raw display $\Q_{508621}$$x$$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}$
\(158113511\) Copy content Toggle raw display $\Q_{158113511}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{158113511}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{158113511}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(1092887040821\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $16$$1$$16$$0$$C_{16}$$[\ ]^{16}$
\(341802543194293\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $19$$1$$19$$0$$C_{19}$$[\ ]^{19}$