Properties

Label 23.5.137...908.1
Degree $23$
Signature $[5, 9]$
Discriminant $-1.377\times 10^{36}$
Root discriminant \(37.26\)
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 - 47*x^9 + 12*x^8 + 43*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 - 47*y^9 + 12*y^8 + 43*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 - 47*x^9 + 12*x^8 + 43*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 - 47*x^9 + 12*x^8 + 43*x^7 + 17*x^6 - 17*x^5 - 16*x^4 + 6*x^2 + x - 1)
 

\( x^{23} - 3 x^{21} - 6 x^{20} + x^{19} + 17 x^{18} + 18 x^{17} - 10 x^{16} - 43 x^{15} - 25 x^{14} + 33 x^{13} + 60 x^{12} + 11 x^{11} - 54 x^{10} - 47 x^{9} + 12 x^{8} + 43 x^{7} + 17 x^{6} - 17 x^{5} + \cdots - 1 \) 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:   \(-1377082951373957855841789612501108908\) \(\medspace = -\,2^{2}\cdot 641\cdot 31357146703\cdot 17127956171402395614949\) 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:  \(37.26\)
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:  $2^{2/3}641^{1/2}31357146703^{1/2}17127956171402395614949^{1/2}\approx 9.314010408753599e+17$
Ramified primes:   \(2\), \(641\), \(31357146703\), \(17127956171402395614949\) 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:  $\Q(\sqrt{-34427\!\cdots\!77227}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:   $a^{22}-3a^{20}-6a^{19}+a^{18}+17a^{17}+18a^{16}-10a^{15}-43a^{14}-25a^{13}+33a^{12}+60a^{11}+11a^{10}-54a^{9}-47a^{8}+12a^{7}+43a^{6}+17a^{5}-17a^{4}-16a^{3}+6a+1$, $a^{22}-3a^{20}-6a^{19}+a^{18}+17a^{17}+18a^{16}-10a^{15}-43a^{14}-25a^{13}+33a^{12}+60a^{11}+11a^{10}-54a^{9}-47a^{8}+12a^{7}+43a^{6}+17a^{5}-16a^{4}-16a^{3}-a^{2}+5a$, $a^{22}-3a^{20}-6a^{19}+a^{18}+17a^{17}+18a^{16}-10a^{15}-43a^{14}-25a^{13}+33a^{12}+60a^{11}+11a^{10}-54a^{9}-47a^{8}+12a^{7}+43a^{6}+18a^{5}-17a^{4}-17a^{3}-2a^{2}+5a+2$, $5a^{22}+a^{21}-14a^{20}-33a^{19}-4a^{18}+80a^{17}+108a^{16}-15a^{15}-207a^{14}-178a^{13}+97a^{12}+308a^{11}+148a^{10}-199a^{9}-278a^{8}-40a^{7}+180a^{6}+138a^{5}-27a^{4}-79a^{3}-30a^{2}+15a+9$, $4a^{22}+a^{21}-11a^{20}-27a^{19}-5a^{18}+63a^{17}+90a^{16}-5a^{15}-164a^{14}-153a^{13}+64a^{12}+248a^{11}+137a^{10}-145a^{9}-231a^{8}-53a^{7}+137a^{6}+122a^{5}-8a^{4}-63a^{3}-30a^{2}+8a+8$, $a^{22}+a^{21}-2a^{20}-9a^{19}-8a^{18}+13a^{17}+35a^{16}+22a^{15}-38a^{14}-71a^{13}-21a^{12}+70a^{11}+81a^{10}-9a^{9}-78a^{8}-52a^{7}+25a^{6}+45a^{5}+13a^{4}-15a^{3}-11a^{2}+1$, $7a^{22}+4a^{21}-19a^{20}-53a^{19}-22a^{18}+108a^{17}+187a^{16}+31a^{15}-287a^{14}-333a^{13}+53a^{12}+451a^{11}+320a^{10}-206a^{9}-440a^{8}-151a^{7}+216a^{6}+234a^{5}+8a^{4}-104a^{3}-57a^{2}+10a+12$, $3a^{22}-2a^{21}-7a^{20}-12a^{19}+10a^{18}+37a^{17}+20a^{16}-38a^{15}-75a^{14}+2a^{13}+82a^{12}+70a^{11}-46a^{10}-97a^{9}-16a^{8}+58a^{7}+46a^{6}-18a^{5}-35a^{4}-4a^{3}+9a^{2}+8a-4$, $4a^{22}+a^{21}-10a^{20}-25a^{19}-7a^{18}+51a^{17}+77a^{16}+8a^{15}-115a^{14}-116a^{13}+23a^{12}+149a^{11}+94a^{10}-67a^{9}-122a^{8}-43a^{7}+49a^{6}+54a^{5}+5a^{4}-18a^{3}-11a^{2}+4a+2$, $4a^{22}+4a^{21}-12a^{20}-35a^{19}-20a^{18}+69a^{17}+134a^{16}+33a^{15}-195a^{14}-254a^{13}+22a^{12}+329a^{11}+259a^{10}-139a^{9}-344a^{8}-129a^{7}+167a^{6}+193a^{5}+12a^{4}-91a^{3}-48a^{2}+9a+13$, $8a^{22}+6a^{21}-21a^{20}-65a^{19}-36a^{18}+122a^{17}+240a^{16}+71a^{15}-337a^{14}-452a^{13}+7a^{12}+563a^{11}+471a^{10}-206a^{9}-587a^{8}-257a^{7}+259a^{6}+331a^{5}+37a^{4}-138a^{3}-83a^{2}+9a+17$, $4a^{22}+2a^{21}-11a^{20}-28a^{19}-11a^{18}+58a^{17}+96a^{16}+15a^{15}-144a^{14}-165a^{13}+23a^{12}+213a^{11}+159a^{10}-87a^{9}-204a^{8}-86a^{7}+84a^{6}+113a^{5}+19a^{4}-44a^{3}-33a^{2}+a+7$, $7a^{22}-5a^{21}-16a^{20}-29a^{19}+26a^{18}+90a^{17}+48a^{16}-100a^{15}-195a^{14}+8a^{13}+225a^{12}+199a^{11}-134a^{10}-282a^{9}-53a^{8}+186a^{7}+155a^{6}-60a^{5}-120a^{4}-14a^{3}+41a^{2}+25a-15$ 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:  \( 3910844212.61 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{9}\cdot 3910844212.61 \cdot 1}{2\cdot\sqrt{1377082951373957855841789612501108908}}\cr\approx \mathstrut & 0.813821695340 \end{aligned}\] (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 - 47*x^9 + 12*x^8 + 43*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 - 47*x^9 + 12*x^8 + 43*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 - 47*x^9 + 12*x^8 + 43*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 - 47*x^9 + 12*x^8 + 43*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 $S_{23}$ 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 ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ $19{,}\,{\href{/padicField/5.4.0.1}{4} }$ $16{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ $19{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ $16{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ $17{,}\,{\href{/padicField/31.6.0.1}{6} }$ $17{,}\,{\href{/padicField/37.6.0.1}{6} }$ $23$ ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ $23$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.9.0.1}{9} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.2.0.1$x^{2} + x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.3.0.1$x^{3} + x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.2.1$x^{3} + 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.0.1$x^{3} + x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.0.1$x^{12} + x^{7} + x^{6} + x^{5} + x^{3} + x + 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(641\) Copy content Toggle raw display $\Q_{641}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $16$$1$$16$$0$$C_{16}$$[\ ]^{16}$
\(31357146703\) Copy content Toggle raw display $\Q_{31357146703}$$x$$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$$[\ ]^{5}$
Deg $11$$1$$11$$0$$C_{11}$$[\ ]^{11}$
\(171\!\cdots\!949\) 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}$