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Properties

Label 23.1.14982213765...2079.1
Degree $23$
Signature $[1, 11]$
Discriminant $-\,1279^{11}$
Root discriminant $30.61$
Ramified prime $1279$
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![-1, 77, -51, -128, 216, 363, -82, 187, 8, -85, -144, -98, -129, 108, -10, 46, 2, 17, -21, 1, -6, 15, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 7*x^22 + 15*x^21 - 6*x^20 + x^19 - 21*x^18 + 17*x^17 + 2*x^16 + 46*x^15 - 10*x^14 + 108*x^13 - 129*x^12 - 98*x^11 - 144*x^10 - 85*x^9 + 8*x^8 + 187*x^7 - 82*x^6 + 363*x^5 + 216*x^4 - 128*x^3 - 51*x^2 + 77*x - 1)
 
gp: K = bnfinit(x^23 - 7*x^22 + 15*x^21 - 6*x^20 + x^19 - 21*x^18 + 17*x^17 + 2*x^16 + 46*x^15 - 10*x^14 + 108*x^13 - 129*x^12 - 98*x^11 - 144*x^10 - 85*x^9 + 8*x^8 + 187*x^7 - 82*x^6 + 363*x^5 + 216*x^4 - 128*x^3 - 51*x^2 + 77*x - 1, 1)
 

Normalized defining polynomial

\( x^{23} - 7 x^{22} + 15 x^{21} - 6 x^{20} + x^{19} - 21 x^{18} + 17 x^{17} + 2 x^{16} + 46 x^{15} - 10 x^{14} + 108 x^{13} - 129 x^{12} - 98 x^{11} - 144 x^{10} - 85 x^{9} + 8 x^{8} + 187 x^{7} - 82 x^{6} + 363 x^{5} + 216 x^{4} - 128 x^{3} - 51 x^{2} + 77 x - 1 \)

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:  \(-14982213765271379362645677088782079=-\,1279^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.61$
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:  $1279$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{117} a^{19} - \frac{4}{117} a^{18} + \frac{1}{39} a^{16} + \frac{2}{39} a^{15} + \frac{1}{39} a^{13} - \frac{1}{39} a^{12} + \frac{10}{117} a^{11} - \frac{19}{117} a^{10} - \frac{2}{13} a^{9} + \frac{1}{13} a^{8} - \frac{14}{39} a^{7} + \frac{1}{13} a^{6} - \frac{16}{39} a^{5} - \frac{17}{39} a^{4} + \frac{43}{117} a^{3} + \frac{5}{117} a^{2} - \frac{16}{39}$, $\frac{1}{1287} a^{20} - \frac{5}{1287} a^{19} - \frac{35}{1287} a^{18} - \frac{4}{143} a^{17} - \frac{49}{1287} a^{16} + \frac{21}{143} a^{15} - \frac{4}{143} a^{14} + \frac{37}{429} a^{13} + \frac{1}{99} a^{12} + \frac{49}{1287} a^{11} - \frac{155}{1287} a^{10} - \frac{10}{143} a^{9} - \frac{103}{1287} a^{8} - \frac{5}{13} a^{7} + \frac{50}{143} a^{6} - \frac{1}{429} a^{5} + \frac{562}{1287} a^{4} - \frac{350}{1287} a^{3} - \frac{161}{1287} a^{2} - \frac{14}{143} a + \frac{152}{1287}$, $\frac{1}{26984529} a^{21} + \frac{311}{2998281} a^{20} + \frac{70535}{26984529} a^{19} + \frac{502259}{26984529} a^{18} - \frac{617366}{26984529} a^{17} - \frac{1471738}{26984529} a^{16} - \frac{114068}{8994843} a^{15} + \frac{543394}{8994843} a^{14} - \frac{51538}{930501} a^{13} - \frac{41321}{691911} a^{12} + \frac{3604760}{26984529} a^{11} - \frac{434743}{26984529} a^{10} + \frac{1142260}{26984529} a^{9} + \frac{988805}{26984529} a^{8} - \frac{3522787}{8994843} a^{7} + \frac{1111949}{8994843} a^{6} - \frac{1076753}{26984529} a^{5} + \frac{2961578}{8994843} a^{4} - \frac{575629}{2075733} a^{3} - \frac{3107410}{26984529} a^{2} - \frac{5536616}{26984529} a + \frac{8057711}{26984529}$, $\frac{1}{229098084534891} a^{22} + \frac{2526244}{229098084534891} a^{21} + \frac{55550659577}{229098084534891} a^{20} - \frac{80868375323}{229098084534891} a^{19} + \frac{205674517250}{76366028178297} a^{18} - \frac{73086666695}{2314122066009} a^{17} + \frac{18310812979}{7899933949479} a^{16} - \frac{12043234646485}{76366028178297} a^{15} - \frac{55426739458}{1602084507237} a^{14} + \frac{4009167573688}{229098084534891} a^{13} + \frac{11533459012970}{229098084534891} a^{12} + \frac{5759332889260}{229098084534891} a^{11} + \frac{11531315387203}{76366028178297} a^{10} - \frac{6825429550696}{76366028178297} a^{9} - \frac{15847214434}{718175813589} a^{8} - \frac{9205913446049}{76366028178297} a^{7} + \frac{12207848623546}{229098084534891} a^{6} - \frac{342925194665}{718175813589} a^{5} + \frac{28024694037935}{229098084534891} a^{4} - \frac{327420771721}{1096163083899} a^{3} - \frac{9743851794725}{25455342726099} a^{2} + \frac{25002102730390}{76366028178297} a + \frac{110096570405135}{229098084534891}$

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:  \( 195904703.063 \) (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 $23$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $23$ $23$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $23$ $23$ $23$ $23$ $23$ $23$

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
1279Data not computed