Properties

Label 23.1.222...471.1
Degree $23$
Signature $[1, 11]$
Discriminant $-2.228\times 10^{42}$
Root discriminant \(69.38\)
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 + 7*x - 4)
 
gp: K = bnfinit(y^23 + 7*y - 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 + 7*x - 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 + 7*x - 4)
 

\( x^{23} + 7x - 4 \) 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:   \(-2227978859786797789863805466564380466715471\) \(\medspace = -\,3\cdot 661\cdot 102898348801\cdot 10918926580157025647925153137\) 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:  \(69.38\)
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:  $3^{1/2}661^{1/2}102898348801^{1/2}10918926580157025647925153137^{1/2}\approx 1.4926415711036585e+21$
Ramified primes:   \(3\), \(661\), \(102898348801\), \(10918926580157025647925153137\) 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:  $\Q(\sqrt{-22279\!\cdots\!15471}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{10}$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{11}$ 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:   $a^{22}+2a^{21}+2a^{20}+a^{19}-a^{18}-\frac{5}{2}a^{17}-3a^{16}-\frac{3}{2}a^{15}+\frac{1}{2}a^{14}+3a^{13}+\frac{7}{2}a^{12}+3a^{11}-3a^{9}-6a^{8}-4a^{7}-\frac{5}{2}a^{6}+5a^{5}+\frac{11}{2}a^{4}+\frac{15}{2}a^{3}+2a^{2}-\frac{5}{2}a-1$, $\frac{1}{2}a^{22}-a^{20}+\frac{1}{2}a^{19}+a^{18}-\frac{3}{2}a^{17}+\frac{1}{2}a^{16}+2a^{15}-\frac{5}{2}a^{14}+2a^{12}-\frac{3}{2}a^{11}-2a^{10}+4a^{9}-\frac{3}{2}a^{8}-a^{7}+\frac{3}{2}a^{6}+\frac{1}{2}a^{5}-2a^{4}-\frac{1}{2}a^{3}+a^{2}+2a-1$, $a^{21}-\frac{1}{2}a^{20}+\frac{3}{2}a^{19}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}+2a^{16}-\frac{3}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{5}{2}a^{12}+a^{11}-4a^{10}-\frac{1}{2}a^{9}+\frac{3}{2}a^{8}-\frac{5}{2}a^{7}+\frac{11}{2}a^{6}-a^{5}+\frac{3}{2}a^{4}+\frac{13}{2}a^{3}-\frac{9}{2}a^{2}+\frac{11}{2}a-3$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+a^{11}+a^{10}-a^{8}-a^{7}-2a^{6}+\frac{3}{2}a^{4}+\frac{5}{2}a^{3}+\frac{3}{2}a^{2}-2a-1$, $2a^{22}+\frac{9}{2}a^{21}+4a^{20}+\frac{1}{2}a^{19}-4a^{18}-\frac{11}{2}a^{17}-\frac{9}{2}a^{16}-\frac{3}{2}a^{15}+\frac{3}{2}a^{14}+5a^{13}+8a^{12}+6a^{11}-\frac{1}{2}a^{10}-10a^{9}-\frac{25}{2}a^{8}-7a^{7}+\frac{5}{2}a^{6}+\frac{19}{2}a^{5}+\frac{21}{2}a^{4}+\frac{21}{2}a^{3}+5a^{2}-4a-5$, $a^{22}+\frac{1}{2}a^{21}+\frac{1}{2}a^{20}+2a^{19}-\frac{3}{2}a^{18}-\frac{3}{2}a^{17}+a^{16}+\frac{5}{2}a^{15}-2a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{10}+\frac{5}{2}a^{9}-3a^{8}-\frac{9}{2}a^{7}+\frac{11}{2}a^{6}+4a^{5}-\frac{9}{2}a^{4}-5a^{3}+\frac{13}{2}a^{2}-\frac{5}{2}a+9$, $a^{22}+\frac{1}{2}a^{21}-a^{20}-a^{19}-\frac{1}{2}a^{18}+\frac{1}{2}a^{17}+\frac{3}{2}a^{16}+\frac{1}{2}a^{15}-a^{14}-\frac{3}{2}a^{13}-\frac{3}{2}a^{12}+2a^{11}+\frac{5}{2}a^{10}+a^{9}-\frac{9}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{2}a^{5}+\frac{5}{2}a^{4}+5a^{3}-\frac{9}{2}a^{2}-\frac{7}{2}a+3$, $a^{22}+a^{21}+a^{20}+\frac{3}{2}a^{19}+\frac{3}{2}a^{18}+a^{17}+a^{16}+\frac{1}{2}a^{15}-\frac{3}{2}a^{14}-3a^{13}-3a^{12}-3a^{11}-2a^{10}+a^{9}+\frac{5}{2}a^{8}+\frac{3}{2}a^{7}+2a^{6}+3a^{5}+\frac{3}{2}a^{4}+\frac{3}{2}a^{3}+3a^{2}+3$, $7a^{22}+\frac{7}{2}a^{21}+3a^{20}+a^{19}+a^{18}-a^{16}-a^{15}-2a^{14}-\frac{1}{2}a^{13}-a^{12}+\frac{1}{2}a^{10}+a^{9}+3a^{8}+2a^{7}+2a^{6}+a^{5}-a^{4}+2a^{3}-\frac{11}{2}a^{2}+a+39$, $13a^{22}+8a^{21}+\frac{5}{2}a^{20}+\frac{5}{2}a^{19}+\frac{3}{2}a^{18}-\frac{3}{2}a^{17}+\frac{3}{2}a^{16}+\frac{1}{2}a^{15}-\frac{5}{2}a^{14}+3a^{13}+a^{12}-3a^{11}+4a^{10}+\frac{1}{2}a^{9}-\frac{7}{2}a^{8}+\frac{9}{2}a^{7}-\frac{5}{2}a^{6}-\frac{9}{2}a^{5}+\frac{13}{2}a^{4}-\frac{11}{2}a^{3}-5a^{2}+11a+85$, $\frac{1}{2}a^{22}+2a^{21}+\frac{7}{2}a^{20}+\frac{3}{2}a^{19}-3a^{18}-\frac{13}{2}a^{17}-4a^{16}+2a^{14}-2a^{13}-\frac{7}{2}a^{12}-\frac{3}{2}a^{11}+6a^{10}+\frac{15}{2}a^{9}+\frac{13}{2}a^{8}+2a^{7}+\frac{5}{2}a^{6}-a^{5}-5a^{4}-10a^{3}-5a^{2}+\frac{3}{2}a+3$ 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:  \( 3129530905130 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 3129530905130 \cdot 1}{2\cdot\sqrt{2227978859786797789863805466564380466715471}}\cr\approx \mathstrut & 1.26328768877246 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^23 + 7*x - 4)
 
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 + 7*x - 4, 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 + 7*x - 4);
 
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 + 7*x - 4);
 
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 $20{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ R $15{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ $22{,}\,{\href{/padicField/7.1.0.1}{1} }$ $19{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ $18{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.13.0.1}{13} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.11.0.1}{11} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $20{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ $19{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.0.1$x^{3} + 2 x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $18$$1$$18$$0$$C_{18}$$[\ ]^{18}$
\(661\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(102898348801\) Copy content Toggle raw display $\Q_{102898348801}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{102898348801}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $9$$1$$9$$0$$C_9$$[\ ]^{9}$
Deg $10$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(109\!\cdots\!137\) Copy content Toggle raw display $\Q_{10\!\cdots\!37}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $20$$1$$20$$0$20T1$[\ ]^{20}$