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Properties

Label 23.1.687...411.1
Degree $23$
Signature $[1, 11]$
Discriminant $-6.872\times 10^{35}$
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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Normalized defining polynomial

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(y^23 - 3*y^22 + 7*y^21 - 8*y^20 - 11*y^19 + 19*y^18 + 160*y^17 + 291*y^16 + 464*y^15 - 35*y^14 - 648*y^13 - 1313*y^12 - 1101*y^11 - 240*y^10 + 1477*y^9 + 1043*y^8 + 863*y^7 - 10*y^6 - 264*y^5 - 712*y^4 - 80*y^3 - 2304*y^2 - 2048*y - 1024, 1)
 
magma: R<x> := PolynomialRing(Rationals()); 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);
 
oscar: Qx, x = PolynomialRing(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)
 

\( x^{23} - 3 x^{22} + 7 x^{21} - 8 x^{20} - 11 x^{19} + 19 x^{18} + 160 x^{17} + 291 x^{16} + 464 x^{15} + \cdots - 1024 \) Copy content Toggle raw display

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

Invariants

Degree:  $23$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-687231335140465279781021461534970411\) \(\medspace = -\,1811^{11}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(36.15\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $1811^{1/2}\approx 42.555845661906424$
Ramified primes:   \(1811\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-1811}) \)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{3}$, $\frac{1}{4}a^{10}-\frac{1}{4}a^{4}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}-\frac{1}{8}a^{5}+\frac{1}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{9}-\frac{1}{8}a^{6}+\frac{1}{8}a^{3}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{10}-\frac{1}{8}a^{7}+\frac{1}{8}a^{4}$, $\frac{1}{32}a^{14}+\frac{1}{32}a^{12}-\frac{3}{32}a^{10}-\frac{1}{32}a^{8}+\frac{1}{8}a^{7}-\frac{1}{32}a^{6}-\frac{1}{8}a^{5}+\frac{3}{32}a^{4}+\frac{3}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{32}a^{15}+\frac{1}{32}a^{13}+\frac{1}{32}a^{11}-\frac{1}{8}a^{10}+\frac{3}{32}a^{9}-\frac{1}{8}a^{8}-\frac{1}{32}a^{7}-\frac{1}{8}a^{6}-\frac{1}{32}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{32}a^{16}+\frac{1}{16}a^{10}-\frac{1}{4}a^{7}-\frac{3}{32}a^{4}+\frac{1}{4}a$, $\frac{1}{64}a^{17}-\frac{1}{64}a^{16}-\frac{1}{64}a^{14}-\frac{1}{16}a^{13}+\frac{3}{64}a^{12}+\frac{1}{32}a^{11}+\frac{5}{64}a^{10}-\frac{1}{16}a^{9}-\frac{7}{64}a^{8}+\frac{1}{8}a^{7}+\frac{13}{64}a^{6}+\frac{1}{64}a^{5}+\frac{3}{16}a^{4}+\frac{1}{8}a^{3}+\frac{3}{8}a^{2}$, $\frac{1}{64}a^{18}-\frac{1}{64}a^{16}-\frac{1}{64}a^{15}-\frac{1}{64}a^{14}-\frac{1}{64}a^{13}+\frac{1}{64}a^{12}-\frac{1}{64}a^{11}-\frac{3}{64}a^{10}+\frac{5}{64}a^{9}-\frac{3}{64}a^{8}+\frac{5}{64}a^{7}-\frac{7}{32}a^{6}+\frac{5}{64}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a^{2}$, $\frac{1}{64}a^{19}-\frac{1}{64}a^{15}-\frac{3}{64}a^{13}-\frac{1}{16}a^{12}-\frac{1}{64}a^{11}-\frac{1}{8}a^{10}+\frac{1}{64}a^{9}-\frac{1}{16}a^{8}-\frac{7}{32}a^{7}-\frac{1}{8}a^{6}-\frac{15}{64}a^{5}-\frac{1}{16}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}$, $\frac{1}{9472}a^{20}-\frac{9}{2368}a^{19}-\frac{7}{4736}a^{18}-\frac{15}{4736}a^{17}-\frac{55}{9472}a^{16}+\frac{7}{1184}a^{15}+\frac{63}{9472}a^{14}-\frac{45}{2368}a^{13}-\frac{155}{9472}a^{12}-\frac{89}{2368}a^{11}-\frac{233}{9472}a^{10}+\frac{99}{2368}a^{9}+\frac{31}{296}a^{8}+\frac{561}{2368}a^{7}-\frac{455}{9472}a^{6}+\frac{215}{4736}a^{5}+\frac{59}{592}a^{4}-\frac{43}{1184}a^{3}+\frac{59}{592}a^{2}-\frac{1}{74}a+\frac{17}{37}$, $\frac{1}{20336384}a^{21}-\frac{73}{10168192}a^{20}+\frac{1825}{10168192}a^{19}+\frac{20291}{10168192}a^{18}+\frac{35953}{20336384}a^{17}+\frac{118271}{10168192}a^{16}-\frac{22969}{20336384}a^{15}+\frac{85615}{10168192}a^{14}+\frac{460981}{20336384}a^{13}-\frac{46043}{10168192}a^{12}+\frac{1179119}{20336384}a^{11}-\frac{140389}{10168192}a^{10}+\frac{23369}{1271024}a^{9}-\frac{42805}{635512}a^{8}-\frac{3774431}{20336384}a^{7}+\frac{45143}{5084096}a^{6}-\frac{345769}{2542048}a^{5}+\frac{409591}{2542048}a^{4}-\frac{50165}{158878}a^{3}-\frac{37089}{317756}a^{2}+\frac{77437}{158878}a-\frac{25291}{79439}$, $\frac{1}{65\!\cdots\!92}a^{22}-\frac{662247085}{32\!\cdots\!96}a^{21}+\frac{5254425991}{65\!\cdots\!92}a^{20}+\frac{80590708193227}{32\!\cdots\!96}a^{19}-\frac{337215451394729}{65\!\cdots\!92}a^{18}-\frac{15399511548717}{40\!\cdots\!12}a^{17}-\frac{1460924573869}{20\!\cdots\!56}a^{16}+\frac{79807531365811}{32\!\cdots\!96}a^{15}+\frac{239445416608027}{16\!\cdots\!48}a^{14}+\frac{48549884800853}{17\!\cdots\!84}a^{13}+\frac{703781746693539}{16\!\cdots\!48}a^{12}-\frac{15\!\cdots\!87}{32\!\cdots\!96}a^{11}+\frac{20440656984429}{34\!\cdots\!68}a^{10}-\frac{52985264500669}{858596971587392}a^{9}-\frac{58\!\cdots\!23}{65\!\cdots\!92}a^{8}+\frac{22\!\cdots\!25}{16\!\cdots\!48}a^{7}+\frac{87\!\cdots\!57}{65\!\cdots\!92}a^{6}-\frac{10\!\cdots\!73}{32\!\cdots\!96}a^{5}+\frac{672725219643027}{40\!\cdots\!12}a^{4}+\frac{819727256289789}{81\!\cdots\!24}a^{3}-\frac{18\!\cdots\!81}{40\!\cdots\!12}a^{2}-\frac{176876261463597}{509791951880014}a-\frac{52392084205566}{254895975940007}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{16049104621149}{81\!\cdots\!24}a^{22}-\frac{483499096368431}{65\!\cdots\!92}a^{21}+\frac{289414792855045}{16\!\cdots\!48}a^{20}-\frac{760126833745943}{32\!\cdots\!96}a^{19}-\frac{556473179543109}{32\!\cdots\!96}a^{18}+\frac{41\!\cdots\!69}{65\!\cdots\!92}a^{17}+\frac{47\!\cdots\!55}{16\!\cdots\!48}a^{16}+\frac{18\!\cdots\!87}{65\!\cdots\!92}a^{15}+\frac{35\!\cdots\!89}{81\!\cdots\!24}a^{14}-\frac{44\!\cdots\!71}{65\!\cdots\!92}a^{13}-\frac{45\!\cdots\!23}{40\!\cdots\!12}a^{12}-\frac{77\!\cdots\!85}{65\!\cdots\!92}a^{11}+\frac{54255115796577}{81\!\cdots\!24}a^{10}+\frac{96\!\cdots\!85}{16\!\cdots\!48}a^{9}+\frac{12\!\cdots\!09}{509791951880014}a^{8}-\frac{21\!\cdots\!55}{65\!\cdots\!92}a^{7}+\frac{39\!\cdots\!31}{32\!\cdots\!96}a^{6}-\frac{53\!\cdots\!25}{16\!\cdots\!48}a^{5}+\frac{10\!\cdots\!99}{40\!\cdots\!12}a^{4}-\frac{46\!\cdots\!49}{40\!\cdots\!12}a^{3}+\frac{358866945431741}{20\!\cdots\!56}a^{2}-\frac{35\!\cdots\!39}{10\!\cdots\!28}a-\frac{119860057563846}{254895975940007}$, $\frac{1626709}{3905656064}a^{22}-\frac{4903109}{3905656064}a^{21}+\frac{2608139}{976414016}a^{20}-\frac{2486541}{976414016}a^{19}-\frac{22976581}{3905656064}a^{18}+\frac{29975019}{3905656064}a^{17}+\frac{294373149}{3905656064}a^{16}+\frac{416155717}{3905656064}a^{15}+\frac{562505147}{3905656064}a^{14}-\frac{218933745}{3905656064}a^{13}-\frac{1169337807}{3905656064}a^{12}-\frac{1928129947}{3905656064}a^{11}-\frac{669273651}{1952828032}a^{10}-\frac{146900351}{976414016}a^{9}+\frac{1945703453}{3905656064}a^{8}+\frac{2202230825}{3905656064}a^{7}+\frac{149662583}{488207008}a^{6}+\frac{102404477}{976414016}a^{5}-\frac{38754557}{488207008}a^{4}-\frac{5872647}{30512938}a^{3}-\frac{31094261}{122051752}a^{2}+\frac{9299}{270026}a-\frac{13249185}{15256469}$, $\frac{36471690905829}{65\!\cdots\!92}a^{22}-\frac{5576555494037}{40\!\cdots\!12}a^{21}+\frac{203548979994139}{65\!\cdots\!92}a^{20}-\frac{148027709873775}{32\!\cdots\!96}a^{19}-\frac{97208699676069}{65\!\cdots\!92}a^{18}-\frac{223915351861999}{32\!\cdots\!96}a^{17}+\frac{17\!\cdots\!89}{16\!\cdots\!48}a^{16}+\frac{19\!\cdots\!65}{81\!\cdots\!24}a^{15}+\frac{25\!\cdots\!13}{81\!\cdots\!24}a^{14}-\frac{21\!\cdots\!73}{16\!\cdots\!48}a^{13}-\frac{28\!\cdots\!27}{40\!\cdots\!12}a^{12}-\frac{19\!\cdots\!23}{16\!\cdots\!48}a^{11}-\frac{28\!\cdots\!57}{65\!\cdots\!92}a^{10}+\frac{49\!\cdots\!73}{81\!\cdots\!24}a^{9}+\frac{95\!\cdots\!45}{65\!\cdots\!92}a^{8}+\frac{28\!\cdots\!69}{32\!\cdots\!96}a^{7}-\frac{22\!\cdots\!19}{65\!\cdots\!92}a^{6}-\frac{22\!\cdots\!31}{32\!\cdots\!96}a^{5}+\frac{142298991877369}{20\!\cdots\!56}a^{4}+\frac{37\!\cdots\!81}{81\!\cdots\!24}a^{3}-\frac{23\!\cdots\!19}{40\!\cdots\!12}a^{2}-\frac{8041331134767}{4511433202478}a-\frac{181736667078104}{254895975940007}$, $\frac{20174724465681}{34\!\cdots\!68}a^{22}-\frac{15\!\cdots\!41}{65\!\cdots\!92}a^{21}+\frac{10\!\cdots\!59}{16\!\cdots\!48}a^{20}-\frac{18\!\cdots\!71}{16\!\cdots\!48}a^{19}+\frac{16\!\cdots\!37}{65\!\cdots\!92}a^{18}+\frac{10\!\cdots\!19}{65\!\cdots\!92}a^{17}+\frac{44\!\cdots\!27}{65\!\cdots\!92}a^{16}+\frac{63\!\cdots\!13}{65\!\cdots\!92}a^{15}+\frac{93\!\cdots\!01}{65\!\cdots\!92}a^{14}-\frac{10\!\cdots\!65}{65\!\cdots\!92}a^{13}-\frac{14\!\cdots\!61}{65\!\cdots\!92}a^{12}-\frac{27\!\cdots\!03}{65\!\cdots\!92}a^{11}-\frac{89\!\cdots\!65}{32\!\cdots\!96}a^{10}+\frac{92\!\cdots\!63}{40\!\cdots\!12}a^{9}+\frac{28\!\cdots\!07}{65\!\cdots\!92}a^{8}+\frac{15\!\cdots\!41}{65\!\cdots\!92}a^{7}+\frac{25\!\cdots\!63}{16\!\cdots\!48}a^{6}+\frac{11\!\cdots\!89}{10\!\cdots\!28}a^{5}-\frac{39\!\cdots\!55}{81\!\cdots\!24}a^{4}+\frac{13\!\cdots\!49}{509791951880014}a^{3}-\frac{25\!\cdots\!91}{509791951880014}a^{2}-\frac{17\!\cdots\!04}{254895975940007}a-\frac{13\!\cdots\!13}{254895975940007}$, $\frac{629406958898803}{32\!\cdots\!96}a^{22}-\frac{27\!\cdots\!03}{32\!\cdots\!96}a^{21}+\frac{59\!\cdots\!03}{32\!\cdots\!96}a^{20}-\frac{16\!\cdots\!03}{81\!\cdots\!24}a^{19}-\frac{94\!\cdots\!99}{32\!\cdots\!96}a^{18}+\frac{31\!\cdots\!11}{32\!\cdots\!96}a^{17}+\frac{12\!\cdots\!43}{40\!\cdots\!12}a^{16}-\frac{33\!\cdots\!83}{32\!\cdots\!96}a^{15}-\frac{751646682229713}{220450573785952}a^{14}-\frac{40\!\cdots\!53}{32\!\cdots\!96}a^{13}-\frac{46\!\cdots\!67}{81\!\cdots\!24}a^{12}+\frac{96\!\cdots\!49}{881802295143808}a^{11}+\frac{69\!\cdots\!57}{32\!\cdots\!96}a^{10}+\frac{80\!\cdots\!25}{16\!\cdots\!48}a^{9}+\frac{43\!\cdots\!69}{32\!\cdots\!96}a^{8}-\frac{82\!\cdots\!71}{32\!\cdots\!96}a^{7}-\frac{30\!\cdots\!45}{32\!\cdots\!96}a^{6}+\frac{14\!\cdots\!83}{81\!\cdots\!24}a^{5}+\frac{49\!\cdots\!15}{509791951880014}a^{4}-\frac{15\!\cdots\!21}{40\!\cdots\!12}a^{3}+\frac{19\!\cdots\!13}{10\!\cdots\!28}a^{2}+\frac{49\!\cdots\!61}{254895975940007}a+\frac{40\!\cdots\!45}{254895975940007}$, $\frac{25239381007209}{16\!\cdots\!48}a^{22}-\frac{121057455315781}{16\!\cdots\!48}a^{21}+\frac{250001826522347}{16\!\cdots\!48}a^{20}-\frac{110823854178065}{81\!\cdots\!24}a^{19}-\frac{300422283606279}{81\!\cdots\!24}a^{18}+\frac{18\!\cdots\!09}{16\!\cdots\!48}a^{17}+\frac{211473285217643}{858596971587392}a^{16}-\frac{17\!\cdots\!69}{81\!\cdots\!24}a^{15}-\frac{10\!\cdots\!23}{16\!\cdots\!48}a^{14}-\frac{10\!\cdots\!89}{10\!\cdots\!28}a^{13}-\frac{848363884158841}{16\!\cdots\!48}a^{12}+\frac{15\!\cdots\!17}{81\!\cdots\!24}a^{11}+\frac{19\!\cdots\!01}{81\!\cdots\!24}a^{10}+\frac{26\!\cdots\!39}{16\!\cdots\!48}a^{9}-\frac{26\!\cdots\!27}{20\!\cdots\!56}a^{8}-\frac{21\!\cdots\!17}{81\!\cdots\!24}a^{7}-\frac{130429824325699}{858596971587392}a^{6}+\frac{33\!\cdots\!93}{16\!\cdots\!48}a^{5}-\frac{15\!\cdots\!29}{40\!\cdots\!12}a^{4}-\frac{897363664557639}{254895975940007}a^{3}+\frac{155774437721109}{107324621448424}a^{2}+\frac{14\!\cdots\!13}{509791951880014}a+\frac{455625475690631}{254895975940007}$, $\frac{1910523217639}{881802295143808}a^{22}-\frac{77799738759033}{16\!\cdots\!48}a^{21}+\frac{32560091213395}{34\!\cdots\!68}a^{20}-\frac{11211803304133}{20\!\cdots\!56}a^{19}-\frac{72999184303007}{20\!\cdots\!56}a^{18}+\frac{406056239360129}{32\!\cdots\!96}a^{17}+\frac{26\!\cdots\!91}{65\!\cdots\!92}a^{16}+\frac{14\!\cdots\!71}{16\!\cdots\!48}a^{15}+\frac{92\!\cdots\!05}{65\!\cdots\!92}a^{14}+\frac{21419514276035}{72182931239648}a^{13}-\frac{14\!\cdots\!97}{65\!\cdots\!92}a^{12}-\frac{40\!\cdots\!55}{81\!\cdots\!24}a^{11}-\frac{30\!\cdots\!77}{65\!\cdots\!92}a^{10}-\frac{18\!\cdots\!59}{16\!\cdots\!48}a^{9}+\frac{20\!\cdots\!05}{32\!\cdots\!96}a^{8}+\frac{14\!\cdots\!75}{16\!\cdots\!48}a^{7}+\frac{40\!\cdots\!45}{65\!\cdots\!92}a^{6}-\frac{45\!\cdots\!87}{32\!\cdots\!96}a^{5}-\frac{74\!\cdots\!03}{10\!\cdots\!28}a^{4}-\frac{78\!\cdots\!35}{81\!\cdots\!24}a^{3}-\frac{17\!\cdots\!87}{40\!\cdots\!12}a^{2}-\frac{170034158460019}{509791951880014}a+\frac{311280193091602}{254895975940007}$, $\frac{63159643144013}{65\!\cdots\!92}a^{22}-\frac{189189347811017}{65\!\cdots\!92}a^{21}+\frac{417595819169411}{65\!\cdots\!92}a^{20}-\frac{117162340057627}{16\!\cdots\!48}a^{19}-\frac{675247197955175}{65\!\cdots\!92}a^{18}+\frac{890409850473957}{65\!\cdots\!92}a^{17}+\frac{27\!\cdots\!87}{16\!\cdots\!48}a^{16}+\frac{17\!\cdots\!85}{65\!\cdots\!92}a^{15}+\frac{312500690835135}{858596971587392}a^{14}-\frac{12\!\cdots\!25}{65\!\cdots\!92}a^{13}-\frac{12\!\cdots\!51}{16\!\cdots\!48}a^{12}-\frac{76\!\cdots\!87}{65\!\cdots\!92}a^{11}-\frac{30\!\cdots\!21}{65\!\cdots\!92}a^{10}+\frac{49\!\cdots\!23}{16\!\cdots\!48}a^{9}+\frac{10\!\cdots\!57}{65\!\cdots\!92}a^{8}+\frac{43\!\cdots\!45}{65\!\cdots\!92}a^{7}-\frac{238323316992089}{65\!\cdots\!92}a^{6}-\frac{49\!\cdots\!59}{32\!\cdots\!96}a^{5}+\frac{992250699022595}{81\!\cdots\!24}a^{4}-\frac{45\!\cdots\!57}{81\!\cdots\!24}a^{3}-\frac{897935594452935}{40\!\cdots\!12}a^{2}-\frac{893540461088041}{509791951880014}a-\frac{351296155357624}{254895975940007}$, $\frac{84582887424471}{65\!\cdots\!92}a^{22}-\frac{198688475982319}{65\!\cdots\!92}a^{21}+\frac{423024348618549}{65\!\cdots\!92}a^{20}-\frac{50183852923415}{16\!\cdots\!48}a^{19}-\frac{14\!\cdots\!93}{65\!\cdots\!92}a^{18}+\frac{12\!\cdots\!91}{65\!\cdots\!92}a^{17}+\frac{901187525606891}{40\!\cdots\!12}a^{16}+\frac{33\!\cdots\!95}{65\!\cdots\!92}a^{15}+\frac{809128085110515}{10\!\cdots\!28}a^{14}+\frac{316940254156481}{577463449917184}a^{13}-\frac{110197139854363}{40\!\cdots\!12}a^{12}-\frac{37\!\cdots\!41}{65\!\cdots\!92}a^{11}-\frac{60\!\cdots\!95}{65\!\cdots\!92}a^{10}-\frac{10\!\cdots\!35}{81\!\cdots\!24}a^{9}-\frac{10\!\cdots\!09}{65\!\cdots\!92}a^{8}-\frac{15\!\cdots\!69}{65\!\cdots\!92}a^{7}-\frac{17\!\cdots\!27}{65\!\cdots\!92}a^{6}-\frac{58\!\cdots\!09}{32\!\cdots\!96}a^{5}-\frac{59\!\cdots\!95}{81\!\cdots\!24}a^{4}-\frac{11\!\cdots\!07}{81\!\cdots\!24}a^{3}-\frac{50\!\cdots\!89}{40\!\cdots\!12}a^{2}-\frac{167007312979405}{254895975940007}a-\frac{22658979317988}{254895975940007}$, $\frac{70544021591959}{16\!\cdots\!48}a^{22}-\frac{30933613763017}{17\!\cdots\!16}a^{21}+\frac{33\!\cdots\!37}{65\!\cdots\!92}a^{20}-\frac{31\!\cdots\!83}{32\!\cdots\!96}a^{19}+\frac{10\!\cdots\!15}{16\!\cdots\!48}a^{18}+\frac{538336265782101}{65\!\cdots\!92}a^{17}+\frac{39\!\cdots\!37}{65\!\cdots\!92}a^{16}+\frac{52\!\cdots\!77}{65\!\cdots\!92}a^{15}+\frac{87\!\cdots\!83}{65\!\cdots\!92}a^{14}-\frac{99\!\cdots\!53}{65\!\cdots\!92}a^{13}-\frac{11\!\cdots\!59}{65\!\cdots\!92}a^{12}-\frac{31\!\cdots\!15}{65\!\cdots\!92}a^{11}-\frac{51\!\cdots\!49}{65\!\cdots\!92}a^{10}+\frac{33\!\cdots\!79}{16\!\cdots\!48}a^{9}+\frac{21\!\cdots\!85}{40\!\cdots\!12}a^{8}+\frac{60\!\cdots\!87}{65\!\cdots\!92}a^{7}+\frac{81\!\cdots\!51}{65\!\cdots\!92}a^{6}-\frac{96\!\cdots\!73}{32\!\cdots\!96}a^{5}-\frac{85\!\cdots\!59}{81\!\cdots\!24}a^{4}+\frac{21\!\cdots\!47}{81\!\cdots\!24}a^{3}-\frac{22\!\cdots\!69}{40\!\cdots\!12}a^{2}-\frac{14\!\cdots\!30}{254895975940007}a-\frac{647062584179066}{254895975940007}$, $\frac{22581079455851}{65\!\cdots\!92}a^{22}-\frac{46577474177481}{32\!\cdots\!96}a^{21}+\frac{379677810469955}{65\!\cdots\!92}a^{20}-\frac{492040360601095}{32\!\cdots\!96}a^{19}+\frac{14\!\cdots\!33}{65\!\cdots\!92}a^{18}-\frac{303717769713757}{16\!\cdots\!48}a^{17}+\frac{943854135412219}{32\!\cdots\!96}a^{16}+\frac{41\!\cdots\!61}{32\!\cdots\!96}a^{15}+\frac{12\!\cdots\!11}{32\!\cdots\!96}a^{14}-\frac{37\!\cdots\!39}{32\!\cdots\!96}a^{13}-\frac{656323481080105}{17\!\cdots\!84}a^{12}-\frac{55\!\cdots\!85}{32\!\cdots\!96}a^{11}-\frac{62\!\cdots\!45}{65\!\cdots\!92}a^{10}+\frac{39\!\cdots\!65}{81\!\cdots\!24}a^{9}+\frac{16\!\cdots\!51}{65\!\cdots\!92}a^{8}+\frac{12\!\cdots\!03}{858596971587392}a^{7}+\frac{12\!\cdots\!01}{65\!\cdots\!92}a^{6}-\frac{57\!\cdots\!37}{32\!\cdots\!96}a^{5}-\frac{634785107748651}{20\!\cdots\!56}a^{4}+\frac{44\!\cdots\!57}{81\!\cdots\!24}a^{3}+\frac{25\!\cdots\!35}{40\!\cdots\!12}a^{2}-\frac{22\!\cdots\!51}{509791951880014}a-\frac{777820099572}{2255716601239}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 9293967260.34 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 9293967260.34 \cdot 1}{2\cdot\sqrt{687231335140465279781021461534970411}}\cr\approx \mathstrut & 6.75504448634 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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 $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

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

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(1811\) Copy content Toggle raw display $\Q_{1811}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$