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Properties

Label 23.1.68723133514...0411.1
Degree $23$
Signature $[1, 11]$
Discriminant $-\,1811^{11}$
Root discriminant $36.15$
Ramified prime $1811$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{23}$ (as 23T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1024, -2048, -2304, -80, -712, -264, -10, 863, 1043, 1477, -240, -1101, -1313, -648, -35, 464, 291, 160, 19, -11, -8, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 3*x^22 + 7*x^21 - 8*x^20 - 11*x^19 + 19*x^18 + 160*x^17 + 291*x^16 + 464*x^15 - 35*x^14 - 648*x^13 - 1313*x^12 - 1101*x^11 - 240*x^10 + 1477*x^9 + 1043*x^8 + 863*x^7 - 10*x^6 - 264*x^5 - 712*x^4 - 80*x^3 - 2304*x^2 - 2048*x - 1024)
 
gp: K = bnfinit(x^23 - 3*x^22 + 7*x^21 - 8*x^20 - 11*x^19 + 19*x^18 + 160*x^17 + 291*x^16 + 464*x^15 - 35*x^14 - 648*x^13 - 1313*x^12 - 1101*x^11 - 240*x^10 + 1477*x^9 + 1043*x^8 + 863*x^7 - 10*x^6 - 264*x^5 - 712*x^4 - 80*x^3 - 2304*x^2 - 2048*x - 1024, 1)
 

Normalized defining polynomial

\( x^{23} - 3 x^{22} + 7 x^{21} - 8 x^{20} - 11 x^{19} + 19 x^{18} + 160 x^{17} + 291 x^{16} + 464 x^{15} - 35 x^{14} - 648 x^{13} - 1313 x^{12} - 1101 x^{11} - 240 x^{10} + 1477 x^{9} + 1043 x^{8} + 863 x^{7} - 10 x^{6} - 264 x^{5} - 712 x^{4} - 80 x^{3} - 2304 x^{2} - 2048 x - 1024 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $23$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-687231335140465279781021461534970411=-\,1811^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.15$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $1811$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{4}$, $\frac{1}{32} a^{14} + \frac{1}{32} a^{12} - \frac{3}{32} a^{10} - \frac{1}{32} a^{8} + \frac{1}{8} a^{7} - \frac{1}{32} a^{6} - \frac{1}{8} a^{5} + \frac{3}{32} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{15} + \frac{1}{32} a^{13} + \frac{1}{32} a^{11} - \frac{1}{8} a^{10} + \frac{3}{32} a^{9} - \frac{1}{8} a^{8} - \frac{1}{32} a^{7} - \frac{1}{8} a^{6} - \frac{1}{32} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{3}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{64} a^{17} - \frac{1}{64} a^{16} - \frac{1}{64} a^{14} - \frac{1}{16} a^{13} + \frac{3}{64} a^{12} + \frac{1}{32} a^{11} + \frac{5}{64} a^{10} - \frac{1}{16} a^{9} - \frac{7}{64} a^{8} + \frac{1}{8} a^{7} + \frac{13}{64} a^{6} + \frac{1}{64} a^{5} + \frac{3}{16} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2}$, $\frac{1}{64} a^{18} - \frac{1}{64} a^{16} - \frac{1}{64} a^{15} - \frac{1}{64} a^{14} - \frac{1}{64} a^{13} + \frac{1}{64} a^{12} - \frac{1}{64} a^{11} - \frac{3}{64} a^{10} + \frac{5}{64} a^{9} - \frac{3}{64} a^{8} + \frac{5}{64} a^{7} - \frac{7}{32} a^{6} + \frac{5}{64} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{64} a^{19} - \frac{1}{64} a^{15} - \frac{3}{64} a^{13} - \frac{1}{16} a^{12} - \frac{1}{64} a^{11} - \frac{1}{8} a^{10} + \frac{1}{64} a^{9} - \frac{1}{16} a^{8} - \frac{7}{32} a^{7} - \frac{1}{8} a^{6} - \frac{15}{64} a^{5} - \frac{1}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2}$, $\frac{1}{9472} a^{20} - \frac{9}{2368} a^{19} - \frac{7}{4736} a^{18} - \frac{15}{4736} a^{17} - \frac{55}{9472} a^{16} + \frac{7}{1184} a^{15} + \frac{63}{9472} a^{14} - \frac{45}{2368} a^{13} - \frac{155}{9472} a^{12} - \frac{89}{2368} a^{11} - \frac{233}{9472} a^{10} + \frac{99}{2368} a^{9} + \frac{31}{296} a^{8} + \frac{561}{2368} a^{7} - \frac{455}{9472} a^{6} + \frac{215}{4736} a^{5} + \frac{59}{592} a^{4} - \frac{43}{1184} a^{3} + \frac{59}{592} a^{2} - \frac{1}{74} a + \frac{17}{37}$, $\frac{1}{20336384} a^{21} - \frac{73}{10168192} a^{20} + \frac{1825}{10168192} a^{19} + \frac{20291}{10168192} a^{18} + \frac{35953}{20336384} a^{17} + \frac{118271}{10168192} a^{16} - \frac{22969}{20336384} a^{15} + \frac{85615}{10168192} a^{14} + \frac{460981}{20336384} a^{13} - \frac{46043}{10168192} a^{12} + \frac{1179119}{20336384} a^{11} - \frac{140389}{10168192} a^{10} + \frac{23369}{1271024} a^{9} - \frac{42805}{635512} a^{8} - \frac{3774431}{20336384} a^{7} + \frac{45143}{5084096} a^{6} - \frac{345769}{2542048} a^{5} + \frac{409591}{2542048} a^{4} - \frac{50165}{158878} a^{3} - \frac{37089}{317756} a^{2} + \frac{77437}{158878} a - \frac{25291}{79439}$, $\frac{1}{65253369840641792} a^{22} - \frac{662247085}{32626684920320896} a^{21} + \frac{5254425991}{65253369840641792} a^{20} + \frac{80590708193227}{32626684920320896} a^{19} - \frac{337215451394729}{65253369840641792} a^{18} - \frac{15399511548717}{4078335615040112} a^{17} - \frac{1460924573869}{2039167807520056} a^{16} + \frac{79807531365811}{32626684920320896} a^{15} + \frac{239445416608027}{16313342460160448} a^{14} + \frac{48549884800853}{1717193943174784} a^{13} + \frac{703781746693539}{16313342460160448} a^{12} - \frac{1543103511103687}{32626684920320896} a^{11} + \frac{20440656984429}{3434387886349568} a^{10} - \frac{52985264500669}{858596971587392} a^{9} - \frac{5853528238691323}{65253369840641792} a^{8} + \frac{2241989413909625}{16313342460160448} a^{7} + \frac{8770234168596357}{65253369840641792} a^{6} - \frac{1059255468919173}{32626684920320896} a^{5} + \frac{672725219643027}{4078335615040112} a^{4} + \frac{819727256289789}{8156671230080224} a^{3} - \frac{1835233526121381}{4078335615040112} a^{2} - \frac{176876261463597}{509791951880014} a - \frac{52392084205566}{254895975940007}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9293967260.34 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{23}$ (as 23T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 46
The 13 conjugacy class representatives for $D_{23}$
Character table for $D_{23}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $23$ $23$ $23$ $23$ $23$ $23$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $23$ $23$ $23$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
1811Data not computed