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Properties

Label 23.23.1260708975...3657.1
Degree $23$
Signature $[23, 0]$
Discriminant $23593^{11}$
Root discriminant $123.40$
Ramified prime $23593$
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![-235248461, 85927891, 1139070319, -1079773672, -1168158873, 1662322180, 357772746, -1107595373, 86961683, 401542378, -92539623, -85904881, 28842666, 11025393, -4813093, -807088, 472463, 26294, -27195, 333, 847, -49, -11, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 11*x^22 - 49*x^21 + 847*x^20 + 333*x^19 - 27195*x^18 + 26294*x^17 + 472463*x^16 - 807088*x^15 - 4813093*x^14 + 11025393*x^13 + 28842666*x^12 - 85904881*x^11 - 92539623*x^10 + 401542378*x^9 + 86961683*x^8 - 1107595373*x^7 + 357772746*x^6 + 1662322180*x^5 - 1168158873*x^4 - 1079773672*x^3 + 1139070319*x^2 + 85927891*x - 235248461)
 
gp: K = bnfinit(x^23 - 11*x^22 - 49*x^21 + 847*x^20 + 333*x^19 - 27195*x^18 + 26294*x^17 + 472463*x^16 - 807088*x^15 - 4813093*x^14 + 11025393*x^13 + 28842666*x^12 - 85904881*x^11 - 92539623*x^10 + 401542378*x^9 + 86961683*x^8 - 1107595373*x^7 + 357772746*x^6 + 1662322180*x^5 - 1168158873*x^4 - 1079773672*x^3 + 1139070319*x^2 + 85927891*x - 235248461, 1)
 

Normalized defining polynomial

\( x^{23} - 11 x^{22} - 49 x^{21} + 847 x^{20} + 333 x^{19} - 27195 x^{18} + 26294 x^{17} + 472463 x^{16} - 807088 x^{15} - 4813093 x^{14} + 11025393 x^{13} + 28842666 x^{12} - 85904881 x^{11} - 92539623 x^{10} + 401542378 x^{9} + 86961683 x^{8} - 1107595373 x^{7} + 357772746 x^{6} + 1662322180 x^{5} - 1168158873 x^{4} - 1079773672 x^{3} + 1139070319 x^{2} + 85927891 x - 235248461 \)

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

Invariants

Degree:  $23$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[23, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1260708975389644854773397594813026107886327713657=23593^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.40$
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:  $23593$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{12} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{12} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{35} a^{17} + \frac{3}{35} a^{16} - \frac{3}{35} a^{15} + \frac{2}{35} a^{14} + \frac{3}{35} a^{13} + \frac{4}{35} a^{12} - \frac{12}{35} a^{11} + \frac{1}{7} a^{10} - \frac{3}{7} a^{9} + \frac{17}{35} a^{8} + \frac{3}{35} a^{7} + \frac{2}{7} a^{6} + \frac{4}{35} a^{5} + \frac{9}{35} a^{4} + \frac{1}{35} a^{3} - \frac{12}{35} a^{2} + \frac{17}{35} a - \frac{1}{5}$, $\frac{1}{175} a^{18} + \frac{1}{175} a^{17} - \frac{9}{175} a^{16} + \frac{1}{175} a^{15} - \frac{8}{175} a^{14} - \frac{2}{175} a^{13} + \frac{22}{175} a^{12} + \frac{36}{175} a^{11} + \frac{24}{175} a^{10} - \frac{13}{35} a^{9} + \frac{12}{35} a^{8} + \frac{11}{175} a^{7} - \frac{72}{175} a^{6} - \frac{6}{175} a^{5} + \frac{4}{175} a^{4} + \frac{2}{5} a^{3} - \frac{8}{175} a^{2} + \frac{22}{175} a - \frac{1}{25}$, $\frac{1}{175} a^{19} + \frac{1}{35} a^{16} - \frac{4}{175} a^{15} - \frac{9}{175} a^{14} - \frac{16}{175} a^{13} + \frac{54}{175} a^{12} + \frac{78}{175} a^{11} + \frac{31}{175} a^{10} + \frac{9}{35} a^{9} - \frac{54}{175} a^{8} - \frac{18}{175} a^{7} + \frac{61}{175} a^{6} - \frac{11}{35} a^{5} + \frac{16}{175} a^{4} - \frac{68}{175} a^{3} + \frac{3}{35} a^{2} - \frac{69}{175} a - \frac{9}{25}$, $\frac{1}{38675} a^{20} + \frac{1}{2275} a^{19} - \frac{57}{38675} a^{18} + \frac{473}{38675} a^{17} + \frac{2694}{38675} a^{16} + \frac{1336}{38675} a^{15} - \frac{479}{5525} a^{14} - \frac{209}{38675} a^{13} + \frac{19307}{38675} a^{12} + \frac{409}{1547} a^{11} + \frac{8409}{38675} a^{10} - \frac{13084}{38675} a^{9} + \frac{19269}{38675} a^{8} - \frac{3777}{38675} a^{7} + \frac{4666}{38675} a^{6} - \frac{6702}{38675} a^{5} + \frac{678}{2275} a^{4} - \frac{783}{5525} a^{3} + \frac{18387}{38675} a^{2} + \frac{1693}{7735} a - \frac{1741}{5525}$, $\frac{1}{143677625} a^{21} - \frac{43}{11052125} a^{20} - \frac{319028}{143677625} a^{19} + \frac{47767}{28735525} a^{18} + \frac{1279456}{143677625} a^{17} + \frac{1951101}{20525375} a^{16} + \frac{398884}{11052125} a^{15} + \frac{2750968}{28735525} a^{14} + \frac{2516354}{28735525} a^{13} + \frac{2626532}{143677625} a^{12} + \frac{27396547}{143677625} a^{11} + \frac{47425523}{143677625} a^{10} + \frac{59661928}{143677625} a^{9} - \frac{41305}{1149421} a^{8} - \frac{834766}{4105075} a^{7} - \frac{31524652}{143677625} a^{6} - \frac{27479182}{143677625} a^{5} + \frac{22579699}{143677625} a^{4} - \frac{100462}{5747105} a^{3} - \frac{15020127}{143677625} a^{2} - \frac{218033}{8451625} a + \frac{7227502}{20525375}$, $\frac{1}{74406008856859361075612185238231424015995909375} a^{22} - \frac{42985230419845334312638743997565951213}{14881201771371872215122437047646284803199181875} a^{21} + \frac{6789212871322115127431249366102095797928}{2010973212347550299340869871303552000432321875} a^{20} - \frac{191793171585373798346441706674099444952636047}{74406008856859361075612185238231424015995909375} a^{19} + \frac{145283419950328810229128670754761615761225496}{74406008856859361075612185238231424015995909375} a^{18} + \frac{471088944799234469390609006422030092098282521}{74406008856859361075612185238231424015995909375} a^{17} + \frac{111681966811826441847133081406094538295814007}{14881201771371872215122437047646284803199181875} a^{16} - \frac{6538770826303656865582375493338419345108175552}{74406008856859361075612185238231424015995909375} a^{15} + \frac{15334332787293204432164180550362953826001992}{2125885967338838887874633863949469257599883125} a^{14} + \frac{525624491568802726133360284728472487850090402}{74406008856859361075612185238231424015995909375} a^{13} + \frac{3990742057196638870398457164537755970213268687}{14881201771371872215122437047646284803199181875} a^{12} + \frac{34903616759406359250962186280515851551655990426}{74406008856859361075612185238231424015995909375} a^{11} + \frac{3291595939215820235174605298270306130391342173}{14881201771371872215122437047646284803199181875} a^{10} + \frac{2918271303681017236938128625964829246569318667}{74406008856859361075612185238231424015995909375} a^{9} - \frac{3826332324754454008769844017300969698337039453}{14881201771371872215122437047646284803199181875} a^{8} - \frac{19345854013287887302036770482827322534493696257}{74406008856859361075612185238231424015995909375} a^{7} - \frac{5161518345584907474569026966304578268743876724}{14881201771371872215122437047646284803199181875} a^{6} - \frac{10392646735124524936526516695217240469145711149}{74406008856859361075612185238231424015995909375} a^{5} - \frac{8289039896838961372229165040930952789471563899}{74406008856859361075612185238231424015995909375} a^{4} - \frac{3198403198876267297767026680514471561223683636}{10629429836694194439373169319747346287999415625} a^{3} - \frac{2541044760254416596064444303807056706801810014}{74406008856859361075612185238231424015995909375} a^{2} - \frac{855418117993715770237254258507319146607019232}{2976240354274374443024487409529256960639836375} a + \frac{934794973503648469114010228956229674490882138}{10629429836694194439373169319747346287999415625}$

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:  $22$
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:  \( 122630019967000000 \) (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 $23$ $23$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $23$ $23$ $23$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
23593Data not computed