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Properties

Label 23.1.69781355284...0671.1
Degree $23$
Signature $[1, 11]$
Discriminant $-\,1471^{11}$
Root discriminant $32.73$
Ramified prime $1471$
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, 107, -9, 358, 249, 291, -275, -383, -521, -211, 240, 274, 180, 147, -89, -68, -56, -31, -6, 43, 3, 3, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 5*x^22 + 3*x^21 + 3*x^20 + 43*x^19 - 6*x^18 - 31*x^17 - 56*x^16 - 68*x^15 - 89*x^14 + 147*x^13 + 180*x^12 + 274*x^11 + 240*x^10 - 211*x^9 - 521*x^8 - 383*x^7 - 275*x^6 + 291*x^5 + 249*x^4 + 358*x^3 - 9*x^2 + 107*x + 1)
 
gp: K = bnfinit(x^23 - 5*x^22 + 3*x^21 + 3*x^20 + 43*x^19 - 6*x^18 - 31*x^17 - 56*x^16 - 68*x^15 - 89*x^14 + 147*x^13 + 180*x^12 + 274*x^11 + 240*x^10 - 211*x^9 - 521*x^8 - 383*x^7 - 275*x^6 + 291*x^5 + 249*x^4 + 358*x^3 - 9*x^2 + 107*x + 1, 1)
 

Normalized defining polynomial

\( x^{23} - 5 x^{22} + 3 x^{21} + 3 x^{20} + 43 x^{19} - 6 x^{18} - 31 x^{17} - 56 x^{16} - 68 x^{15} - 89 x^{14} + 147 x^{13} + 180 x^{12} + 274 x^{11} + 240 x^{10} - 211 x^{9} - 521 x^{8} - 383 x^{7} - 275 x^{6} + 291 x^{5} + 249 x^{4} + 358 x^{3} - 9 x^{2} + 107 x + 1 \)

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:  \(-69781355284066743169711357773860671=-\,1471^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.73$
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:  $1471$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{189} a^{19} + \frac{5}{189} a^{18} + \frac{1}{27} a^{17} - \frac{10}{189} a^{16} - \frac{10}{63} a^{15} + \frac{1}{21} a^{14} - \frac{1}{63} a^{13} - \frac{5}{189} a^{11} - \frac{22}{189} a^{10} - \frac{8}{189} a^{9} - \frac{1}{189} a^{8} + \frac{1}{9} a^{7} - \frac{1}{3} a^{6} + \frac{4}{9} a^{5} + \frac{2}{7} a^{4} + \frac{31}{189} a^{3} - \frac{55}{189} a^{2} - \frac{89}{189} a - \frac{34}{189}$, $\frac{1}{189} a^{20} + \frac{1}{63} a^{18} - \frac{1}{63} a^{17} - \frac{1}{189} a^{16} - \frac{10}{63} a^{15} + \frac{5}{63} a^{14} + \frac{5}{63} a^{13} - \frac{5}{189} a^{12} + \frac{1}{63} a^{11} - \frac{1}{63} a^{10} + \frac{2}{21} a^{9} + \frac{5}{189} a^{8} + \frac{1}{9} a^{7} - \frac{2}{9} a^{6} + \frac{4}{63} a^{5} - \frac{50}{189} a^{4} - \frac{1}{9} a^{3} + \frac{3}{7} a^{2} + \frac{4}{63} a + \frac{23}{189}$, $\frac{1}{334341} a^{21} - \frac{28}{15921} a^{20} + \frac{187}{334341} a^{19} + \frac{566}{47763} a^{18} + \frac{5378}{111447} a^{17} + \frac{9659}{334341} a^{16} + \frac{9946}{111447} a^{15} + \frac{3455}{111447} a^{14} + \frac{40393}{334341} a^{13} - \frac{5273}{37149} a^{12} + \frac{35680}{334341} a^{11} + \frac{10859}{334341} a^{10} - \frac{5662}{111447} a^{9} + \frac{46898}{334341} a^{8} - \frac{3184}{15921} a^{7} + \frac{54226}{111447} a^{6} - \frac{101900}{334341} a^{5} - \frac{26480}{111447} a^{4} - \frac{71831}{334341} a^{3} + \frac{8819}{47763} a^{2} - \frac{9133}{111447} a + \frac{55736}{334341}$, $\frac{1}{1051883319011665959} a^{22} - \frac{1239853753063}{1051883319011665959} a^{21} + \frac{2451181648289930}{1051883319011665959} a^{20} - \frac{1209540197155618}{1051883319011665959} a^{19} + \frac{15408652922852275}{350627773003888653} a^{18} - \frac{440673393684886}{50089681857698379} a^{17} - \frac{11192360177618827}{1051883319011665959} a^{16} + \frac{6601563789704666}{50089681857698379} a^{15} + \frac{59460982130351080}{1051883319011665959} a^{14} - \frac{35367411024375286}{1051883319011665959} a^{13} - \frac{135837130896276700}{1051883319011665959} a^{12} - \frac{134903839943941}{1051883319011665959} a^{11} - \frac{41801438996131145}{350627773003888653} a^{10} + \frac{2272241741346898}{38958641444876517} a^{9} - \frac{94565496507127852}{1051883319011665959} a^{8} - \frac{130675170973421021}{350627773003888653} a^{7} + \frac{38557081525522837}{80914101462435843} a^{6} - \frac{383413186335015436}{1051883319011665959} a^{5} - \frac{458992933261208761}{1051883319011665959} a^{4} - \frac{43280301324506905}{1051883319011665959} a^{3} + \frac{138468885933215323}{350627773003888653} a^{2} - \frac{81957629112798380}{350627773003888653} a - \frac{473502647293008379}{1051883319011665959}$

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:  \( 492297100.302 \) (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 $23$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $23$ $23$ $23$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $23$ $23$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $23$ $23$

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
1471Data not computed