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Properties

Label 23.1.83166243861...5703.1
Degree $23$
Signature $[1, 11]$
Discriminant $-\,647^{11}$
Root discriminant $22.10$
Ramified prime $647$
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, 20, 3, -37, 46, 2, -151, 384, -610, 849, -995, 955, -798, 596, -378, 202, -75, 0, 21, -19, 13, -3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 2*x^22 - 3*x^21 + 13*x^20 - 19*x^19 + 21*x^18 - 75*x^16 + 202*x^15 - 378*x^14 + 596*x^13 - 798*x^12 + 955*x^11 - 995*x^10 + 849*x^9 - 610*x^8 + 384*x^7 - 151*x^6 + 2*x^5 + 46*x^4 - 37*x^3 + 3*x^2 + 20*x + 1)
 
gp: K = bnfinit(x^23 - 2*x^22 - 3*x^21 + 13*x^20 - 19*x^19 + 21*x^18 - 75*x^16 + 202*x^15 - 378*x^14 + 596*x^13 - 798*x^12 + 955*x^11 - 995*x^10 + 849*x^9 - 610*x^8 + 384*x^7 - 151*x^6 + 2*x^5 + 46*x^4 - 37*x^3 + 3*x^2 + 20*x + 1, 1)
 

Normalized defining polynomial

\( x^{23} - 2 x^{22} - 3 x^{21} + 13 x^{20} - 19 x^{19} + 21 x^{18} - 75 x^{16} + 202 x^{15} - 378 x^{14} + 596 x^{13} - 798 x^{12} + 955 x^{11} - 995 x^{10} + 849 x^{9} - 610 x^{8} + 384 x^{7} - 151 x^{6} + 2 x^{5} + 46 x^{4} - 37 x^{3} + 3 x^{2} + 20 x + 1 \)

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:  \(-8316624386164136054632388935703=-\,647^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.10$
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:  $647$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{14} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{19} + \frac{1}{5} a^{14} + \frac{1}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{275} a^{20} - \frac{3}{55} a^{19} + \frac{1}{25} a^{18} + \frac{6}{275} a^{17} + \frac{2}{25} a^{16} - \frac{4}{55} a^{15} + \frac{54}{275} a^{14} + \frac{1}{55} a^{13} - \frac{12}{25} a^{12} + \frac{42}{275} a^{11} + \frac{47}{275} a^{10} + \frac{42}{275} a^{9} - \frac{91}{275} a^{8} + \frac{73}{275} a^{7} - \frac{97}{275} a^{6} - \frac{73}{275} a^{5} + \frac{23}{275} a^{4} + \frac{26}{275} a^{3} + \frac{68}{275} a^{2} + \frac{39}{275} a + \frac{119}{275}$, $\frac{1}{193325} a^{21} + \frac{6}{17575} a^{20} - \frac{19079}{193325} a^{19} - \frac{8948}{193325} a^{18} - \frac{15937}{193325} a^{17} + \frac{9352}{193325} a^{16} + \frac{14164}{193325} a^{15} + \frac{93754}{193325} a^{14} - \frac{4952}{193325} a^{13} - \frac{5474}{38665} a^{12} + \frac{60099}{193325} a^{11} + \frac{52}{5225} a^{10} - \frac{8509}{17575} a^{9} + \frac{55512}{193325} a^{8} - \frac{17779}{193325} a^{7} - \frac{1212}{38665} a^{6} - \frac{12332}{38665} a^{5} + \frac{71629}{193325} a^{4} + \frac{47549}{193325} a^{3} + \frac{3233}{10175} a^{2} - \frac{187}{17575} a + \frac{15414}{193325}$, $\frac{1}{16889065325} a^{22} + \frac{6236}{16889065325} a^{21} + \frac{15050612}{16889065325} a^{20} + \frac{25039191}{456461225} a^{19} + \frac{1541886859}{16889065325} a^{18} + \frac{233319718}{16889065325} a^{17} + \frac{7588288}{456461225} a^{16} + \frac{934312404}{16889065325} a^{15} - \frac{4959075178}{16889065325} a^{14} - \frac{78682011}{3377813065} a^{13} + \frac{536483197}{16889065325} a^{12} + \frac{689341351}{16889065325} a^{11} + \frac{4786760163}{16889065325} a^{10} + \frac{148378891}{888898175} a^{9} + \frac{644631076}{3377813065} a^{8} + \frac{8099028858}{16889065325} a^{7} + \frac{5025800893}{16889065325} a^{6} + \frac{5105693096}{16889065325} a^{5} + \frac{1067482887}{16889065325} a^{4} - \frac{2095142937}{16889065325} a^{3} - \frac{4610451389}{16889065325} a^{2} - \frac{6103940692}{16889065325} a - \frac{3711736931}{16889065325}$

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:  \( 1561186.27544 \) (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 $23$ $23$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $23$ $23$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $23$ $23$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $23$ $23$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
647Data not computed