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Properties

Label 23.7.484...792.1
Degree $23$
Signature $[7, 8]$
Discriminant $4.842\times 10^{33}$
Root discriminant \(29.15\)
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 + 11*x^11 - 54*x^10 - 48*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 + 11*y^11 - 54*y^10 - 48*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 + 11*x^11 - 54*x^10 - 48*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 + 11*x^11 - 54*x^10 - 48*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:  $[7, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(4842220039012226302771299504642792\) \(\medspace = 2^{3}\cdot 3^{2}\cdot 4231\cdot 4210631\cdot 3775042239237949162501\) 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:  \(29.15\)
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:  $2^{3/2}3^{1/2}4231^{1/2}4210631^{1/2}3775042239237949162501^{1/2}\approx 4.017553168705311e+16$
Ramified primes:   \(2\), \(3\), \(4231\), \(4210631\), \(3775042239237949162501\) 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:  $\Q(\sqrt{13450\!\cdots\!84522}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:  $14$
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:   $a$, $a^{4}-a^{2}-a$, $a^{21}-3a^{19}-6a^{18}+a^{17}+17a^{16}+18a^{15}-10a^{14}-43a^{13}-25a^{12}+33a^{11}+60a^{10}+11a^{9}-54a^{8}-48a^{7}+12a^{6}+44a^{5}+17a^{4}-16a^{3}-16a^{2}-a+5$, $5a^{22}-13a^{20}-30a^{19}-a^{18}+73a^{17}+92a^{16}-17a^{15}-179a^{14}-143a^{13}+84a^{12}+251a^{11}+111a^{10}-162a^{9}-217a^{8}-28a^{7}+138a^{6}+99a^{5}-19a^{4}-53a^{3}-19a^{2}+11a+4$, $42a^{22}+21a^{21}-114a^{20}-309a^{19}-117a^{18}+647a^{17}+1081a^{16}+144a^{15}-1709a^{14}-1918a^{13}+371a^{12}+2673a^{11}+1838a^{10}-1275a^{9}-2639a^{8}-876a^{7}+1354a^{6}+1401a^{5}+32a^{4}-638a^{3}-332a^{2}+71a+77$, $3a^{21}-9a^{19}-18a^{18}+3a^{17}+50a^{16}+54a^{15}-28a^{14}-124a^{13}-74a^{12}+89a^{11}+168a^{10}+34a^{9}-141a^{8}-130a^{7}+25a^{6}+107a^{5}+43a^{4}-34a^{3}-33a^{2}-a+9$, $8a^{22}-21a^{20}-48a^{19}+118a^{17}+145a^{16}-35a^{15}-291a^{14}-218a^{13}+152a^{12}+405a^{11}+156a^{10}-278a^{9}-339a^{8}-20a^{7}+230a^{6}+144a^{5}-44a^{4}-83a^{3}-25a^{2}+18a+3$, $5a^{22}+a^{21}-14a^{20}-33a^{19}-4a^{18}+80a^{17}+108a^{16}-15a^{15}-207a^{14}-178a^{13}+97a^{12}+308a^{11}+148a^{10}-199a^{9}-283a^{8}-41a^{7}+184a^{6}+140a^{5}-26a^{4}-80a^{3}-31a^{2}+15a+10$, $17a^{22}+8a^{21}-47a^{20}-125a^{19}-42a^{18}+270a^{17}+437a^{16}+37a^{15}-721a^{14}-774a^{13}+197a^{12}+1131a^{11}+731a^{10}-582a^{9}-1111a^{8}-325a^{7}+609a^{6}+588a^{5}-13a^{4}-286a^{3}-137a^{2}+38a+35$, $a^{22}-3a^{20}-6a^{19}+a^{18}+17a^{17}+18a^{16}-10a^{15}-43a^{14}-25a^{13}+33a^{12}+60a^{11}+11a^{10}-54a^{9}-48a^{8}+12a^{7}+44a^{6}+17a^{5}-17a^{4}-16a^{3}+7a+2$, $21a^{22}+11a^{21}-58a^{20}-156a^{19}-59a^{18}+330a^{17}+549a^{16}+68a^{15}-877a^{14}-978a^{13}+203a^{12}+1376a^{11}+938a^{10}-670a^{9}-1360a^{8}-441a^{7}+711a^{6}+725a^{5}+10a^{4}-334a^{3}-173a^{2}+39a+40$, $14a^{22}+8a^{21}-39a^{20}-106a^{19}-42a^{18}+222a^{17}+376a^{16}+51a^{15}-593a^{14}-671a^{13}+131a^{12}+934a^{11}+640a^{10}-449a^{9}-922a^{8}-297a^{7}+479a^{6}+485a^{5}+4a^{4}-222a^{3}-110a^{2}+26a+26$, $8a^{22}+4a^{21}-23a^{20}-58a^{19}-19a^{18}+128a^{17}+202a^{16}+10a^{15}-334a^{14}-349a^{13}+105a^{12}+512a^{11}+314a^{10}-276a^{9}-490a^{8}-124a^{7}+271a^{6}+248a^{5}-15a^{4}-120a^{3}-53a^{2}+17a+13$, $13a^{22}+3a^{21}-35a^{20}-87a^{19}-16a^{18}+200a^{17}+286a^{16}-14a^{15}-516a^{14}-477a^{13}+200a^{12}+769a^{11}+412a^{10}-454a^{9}-709a^{8}-144a^{7}+427a^{6}+347a^{5}-42a^{4}-180a^{3}-73a^{2}+31a+16$ 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:  \( 179256140.602 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{8}\cdot 179256140.602 \cdot 1}{2\cdot\sqrt{4842220039012226302771299504642792}}\cr\approx \mathstrut & 0.400470658884 \end{aligned}\] (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 + 11*x^11 - 54*x^10 - 48*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 + 11*x^11 - 54*x^10 - 48*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 + 11*x^11 - 54*x^10 - 48*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 + 11*x^11 - 54*x^10 - 48*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 R $20{,}\,{\href{/padicField/5.3.0.1}{3} }$ $17{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.13.0.1}{13} }{,}\,{\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ $23$ ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ $15{,}\,{\href{/padicField/23.8.0.1}{8} }$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ $19{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ $20{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }$ $17{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(2\) Copy content Toggle raw display 2.2.3.2$x^{2} + 4 x + 10$$2$$1$$3$$C_2$$[3]$
2.4.0.1$x^{4} + x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x^{2} + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
2.9.0.1$x^{9} + x^{4} + 1$$1$$9$$0$$C_9$$[\ ]^{9}$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.5.0.1$x^{5} + 2 x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.7.0.1$x^{7} + 2 x^{2} + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
3.7.0.1$x^{7} + 2 x^{2} + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
\(4231\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $17$$1$$17$$0$$C_{17}$$[\ ]^{17}$
\(4210631\) Copy content Toggle raw display $\Q_{4210631}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $11$$1$$11$$0$$C_{11}$$[\ ]^{11}$
\(377\!\cdots\!501\) Copy content Toggle raw display 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 $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$