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Properties

Label 23.7.437...528.1
Degree $23$
Signature $[7, 8]$
Discriminant $4.371\times 10^{38}$
Root discriminant \(47.87\)
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^22 + 4*x^20 + 5*x^19 - 5*x^18 - 14*x^17 + 4*x^16 + 15*x^15 + 10*x^14 - 22*x^13 - 13*x^12 + 16*x^11 + 18*x^10 - 3*x^9 - 23*x^8 + x^7 + 13*x^6 + 4*x^5 - 7*x^4 - 4*x^3 + 5*x^2 + x - 1)
 
gp: K = bnfinit(y^23 - 3*y^22 + 4*y^20 + 5*y^19 - 5*y^18 - 14*y^17 + 4*y^16 + 15*y^15 + 10*y^14 - 22*y^13 - 13*y^12 + 16*y^11 + 18*y^10 - 3*y^9 - 23*y^8 + y^7 + 13*y^6 + 4*y^5 - 7*y^4 - 4*y^3 + 5*y^2 + y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 3*x^22 + 4*x^20 + 5*x^19 - 5*x^18 - 14*x^17 + 4*x^16 + 15*x^15 + 10*x^14 - 22*x^13 - 13*x^12 + 16*x^11 + 18*x^10 - 3*x^9 - 23*x^8 + x^7 + 13*x^6 + 4*x^5 - 7*x^4 - 4*x^3 + 5*x^2 + x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 3*x^22 + 4*x^20 + 5*x^19 - 5*x^18 - 14*x^17 + 4*x^16 + 15*x^15 + 10*x^14 - 22*x^13 - 13*x^12 + 16*x^11 + 18*x^10 - 3*x^9 - 23*x^8 + x^7 + 13*x^6 + 4*x^5 - 7*x^4 - 4*x^3 + 5*x^2 + x - 1)
 

\( x^{23} - 3 x^{22} + 4 x^{20} + 5 x^{19} - 5 x^{18} - 14 x^{17} + 4 x^{16} + 15 x^{15} + 10 x^{14} + \cdots - 1 \) 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:   \(437137538929872152263330629707930008528\) \(\medspace = 2^{4}\cdot 17\cdot 241\cdot 1709\cdot 3877\cdot 111868093\cdot 346250347\cdot 25983511363\) 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:  \(47.87\)
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:  not computed
Ramified primes:   \(2\), \(17\), \(241\), \(1709\), \(3877\), \(111868093\), \(346250347\), \(25983511363\) 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:  $\Q(\sqrt{27321\!\cdots\!25533}$)
$\card{ \Aut(K/\Q) }$:  $1$
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:   $a$, $a^{22}-3a^{21}+4a^{19}+5a^{18}-5a^{17}-14a^{16}+4a^{15}+15a^{14}+10a^{13}-22a^{12}-13a^{11}+16a^{10}+18a^{9}-3a^{8}-23a^{7}+a^{6}+12a^{5}+5a^{4}-6a^{3}-4a^{2}+4a+1$, $3a^{22}-7a^{21}-4a^{20}+7a^{19}+21a^{18}-38a^{16}-18a^{15}+27a^{14}+50a^{13}-26a^{12}-50a^{11}+a^{10}+54a^{9}+30a^{8}-37a^{7}-26a^{6}+14a^{5}+21a^{4}-4a^{3}-13a^{2}+2a+4$, $a^{22}-4a^{21}+3a^{20}+3a^{19}+4a^{18}-11a^{17}-10a^{16}+13a^{15}+12a^{14}+4a^{13}-30a^{12}+6a^{11}+14a^{10}+11a^{9}-15a^{8}-15a^{7}+14a^{6}+2a^{5}-a^{4}-9a^{3}+3a^{2}+2a-2$, $a^{22}-4a^{21}+4a^{20}+a^{19}+2a^{18}-9a^{17}-3a^{16}+14a^{15}+2a^{14}-3a^{13}-24a^{12}+18a^{11}+9a^{10}-15a^{8}-4a^{7}+18a^{6}+a^{5}-3a^{4}-4a^{3}+6a^{2}+2a-2$, $4a^{22}-16a^{21}+16a^{20}+a^{19}+16a^{18}-36a^{17}-17a^{16}+40a^{15}+16a^{14}+10a^{13}-100a^{12}+60a^{11}+21a^{10}+37a^{9}-62a^{8}-29a^{7}+52a^{6}+6a^{5}-4a^{4}-32a^{3}+17a^{2}+9a-6$, $a^{22}-5a^{21}+6a^{20}+4a^{19}-a^{18}-19a^{17}-7a^{16}+35a^{15}+19a^{14}-17a^{13}-61a^{12}+18a^{11}+51a^{10}+15a^{9}-45a^{8}-41a^{7}+37a^{6}+30a^{5}-2a^{4}-25a^{3}-4a^{2}+9a-1$, $2a^{22}-8a^{21}+8a^{20}+a^{19}+6a^{18}-15a^{17}-11a^{16}+22a^{15}+6a^{14}+6a^{13}-47a^{12}+25a^{11}+13a^{10}+12a^{9}-19a^{8}-22a^{7}+27a^{6}+a^{5}-11a^{3}+5a^{2}+3a-1$, $a^{21}-2a^{20}-a^{19}-a^{18}+8a^{17}+3a^{16}-7a^{15}-10a^{14}+16a^{12}-2a^{11}-9a^{10}-12a^{9}+14a^{8}+13a^{7}-4a^{5}-a^{4}+7a^{3}+2a^{2}-a-1$, $2a^{22}-5a^{21}-2a^{20}+6a^{19}+12a^{18}-3a^{17}-26a^{16}-4a^{15}+22a^{14}+27a^{13}-27a^{12}-31a^{11}+14a^{10}+34a^{9}+10a^{8}-33a^{7}-8a^{6}+17a^{5}+12a^{4}-6a^{3}-7a^{2}+7a+4$, $a^{22}-2a^{21}-3a^{20}+5a^{19}+6a^{18}-15a^{16}-5a^{15}+14a^{14}+12a^{13}-10a^{12}-21a^{11}+14a^{10}+16a^{9}+3a^{8}-16a^{7}-6a^{6}+14a^{5}+2a^{4}-7a^{3}-3a^{2}+5a+3$, $2a^{22}-4a^{21}-6a^{20}+9a^{19}+14a^{18}+3a^{17}-36a^{16}-14a^{15}+28a^{14}+39a^{13}-17a^{12}-57a^{11}+19a^{10}+40a^{9}+28a^{8}-48a^{7}-24a^{6}+23a^{5}+19a^{4}-3a^{3}-17a^{2}+7a+6$, $3a^{22}-13a^{21}+15a^{20}+10a^{18}-31a^{17}-12a^{16}+42a^{15}+8a^{14}+4a^{13}-86a^{12}+60a^{11}+21a^{10}+18a^{9}-53a^{8}-29a^{7}+60a^{6}-6a^{5}-2a^{4}-29a^{3}+20a^{2}+9a-8$, $3a^{22}-11a^{21}+7a^{20}+8a^{19}+11a^{18}-23a^{17}-34a^{16}+35a^{15}+36a^{14}+12a^{13}-88a^{12}-a^{11}+60a^{10}+40a^{9}-37a^{8}-71a^{7}+41a^{6}+34a^{5}+a^{4}-31a^{3}-a^{2}+17a-5$ 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:  \( 78508567076.1 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{8}\cdot 78508567076.1 \cdot 1}{2\cdot\sqrt{437137538929872152263330629707930008528}}\cr\approx \mathstrut & 0.583750035762 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^23 - 3*x^22 + 4*x^20 + 5*x^19 - 5*x^18 - 14*x^17 + 4*x^16 + 15*x^15 + 10*x^14 - 22*x^13 - 13*x^12 + 16*x^11 + 18*x^10 - 3*x^9 - 23*x^8 + x^7 + 13*x^6 + 4*x^5 - 7*x^4 - 4*x^3 + 5*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^22 + 4*x^20 + 5*x^19 - 5*x^18 - 14*x^17 + 4*x^16 + 15*x^15 + 10*x^14 - 22*x^13 - 13*x^12 + 16*x^11 + 18*x^10 - 3*x^9 - 23*x^8 + x^7 + 13*x^6 + 4*x^5 - 7*x^4 - 4*x^3 + 5*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^22 + 4*x^20 + 5*x^19 - 5*x^18 - 14*x^17 + 4*x^16 + 15*x^15 + 10*x^14 - 22*x^13 - 13*x^12 + 16*x^11 + 18*x^10 - 3*x^9 - 23*x^8 + x^7 + 13*x^6 + 4*x^5 - 7*x^4 - 4*x^3 + 5*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^22 + 4*x^20 + 5*x^19 - 5*x^18 - 14*x^17 + 4*x^16 + 15*x^15 + 10*x^14 - 22*x^13 - 13*x^12 + 16*x^11 + 18*x^10 - 3*x^9 - 23*x^8 + x^7 + 13*x^6 + 4*x^5 - 7*x^4 - 4*x^3 + 5*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 $17{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.9.0.1}{9} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ $18{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.9.0.1}{9} }$ $18{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ $20{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.11.0.1}{11} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $15{,}\,{\href{/padicField/37.8.0.1}{8} }$ ${\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ $17{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ $20{,}\,{\href{/padicField/59.3.0.1}{3} }$

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 2.4.4.5$x^{4} + 2 x + 2$$4$$1$$4$$S_4$$[4/3, 4/3]_{3}^{2}$
2.19.0.1$x^{19} + x^{5} + x^{2} + x + 1$$1$$19$$0$$C_{19}$$[\ ]^{19}$
\(17\) Copy content Toggle raw display 17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.5.0.1$x^{5} + x + 14$$1$$5$$0$$C_5$$[\ ]^{5}$
17.14.0.1$x^{14} + x^{8} + 11 x^{7} + x^{6} + 8 x^{5} + 16 x^{4} + 13 x^{3} + 9 x^{2} + 3 x + 3$$1$$14$$0$$C_{14}$$[\ ]^{14}$
\(241\) Copy content Toggle raw display $\Q_{241}$$x$$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}$
\(1709\) Copy content Toggle raw display $\Q_{1709}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1709}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{1709}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
\(3877\) Copy content Toggle raw display $\Q_{3877}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $20$$1$$20$$0$20T1$[\ ]^{20}$
\(111868093\) Copy content Toggle raw display $\Q_{111868093}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $16$$1$$16$$0$$C_{16}$$[\ ]^{16}$
\(346250347\) 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}$
\(25983511363\) Copy content Toggle raw display $\Q_{25983511363}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{25983511363}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $15$$1$$15$$0$$C_{15}$$[\ ]^{15}$