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Properties

Label 23.1.149...079.1
Degree $23$
Signature $[1, 11]$
Discriminant $-1.498\times 10^{34}$
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: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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(y^23 - 7*y^22 + 15*y^21 - 6*y^20 + y^19 - 21*y^18 + 17*y^17 + 2*y^16 + 46*y^15 - 10*y^14 + 108*y^13 - 129*y^12 - 98*y^11 - 144*y^10 - 85*y^9 + 8*y^8 + 187*y^7 - 82*y^6 + 363*y^5 + 216*y^4 - 128*y^3 - 51*y^2 + 77*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); 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);
 
oscar: Qx, x = PolynomialRing(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)
 

\( x^{23} - 7 x^{22} + 15 x^{21} - 6 x^{20} + x^{19} - 21 x^{18} + 17 x^{17} + 2 x^{16} + 46 x^{15} + \cdots - 1 \) 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:   \(-14982213765271379362645677088782079\) \(\medspace = -\,1279^{11}\) 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:  \(30.61\)
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:  $1279^{1/2}\approx 35.76310948449533$
Ramified primes:   \(1279\) 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:  \(\Q(\sqrt{-1279}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

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

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:   $\frac{1730550511289}{229098084534891}a^{22}-\frac{13500436210384}{229098084534891}a^{21}+\frac{35349777056872}{229098084534891}a^{20}-\frac{30266212733935}{229098084534891}a^{19}+\frac{335873324104}{5874309859869}a^{18}-\frac{1762948521842}{8485114242033}a^{17}+\frac{389244108026}{1602084507237}a^{16}-\frac{8741128818803}{76366028178297}a^{15}+\frac{102999603146246}{229098084534891}a^{14}-\frac{63982277728315}{229098084534891}a^{13}+\frac{6223692229754}{7899933949479}a^{12}-\frac{459336872375614}{229098084534891}a^{11}-\frac{37616086711351}{76366028178297}a^{10}-\frac{72794110336139}{76366028178297}a^{9}+\frac{175921440822304}{229098084534891}a^{8}+\frac{142835156810789}{76366028178297}a^{7}+\frac{768682488749921}{229098084534891}a^{6}-\frac{94142946573133}{229098084534891}a^{5}+\frac{629449008989497}{229098084534891}a^{4}-\frac{1924712469430}{927522609453}a^{3}-\frac{30915257866274}{8485114242033}a^{2}-\frac{3020493491896}{5874309859869}a+\frac{225867017033689}{229098084534891}$, $\frac{1990066947256}{229098084534891}a^{22}-\frac{948040686833}{17622929579607}a^{21}+\frac{1713177464767}{20827098594081}a^{20}+\frac{11533808706022}{229098084534891}a^{19}-\frac{3667904618308}{76366028178297}a^{18}-\frac{232540695377}{1958103286623}a^{17}-\frac{8611416560926}{229098084534891}a^{16}+\frac{12891738863342}{76366028178297}a^{15}+\frac{5623184344870}{17622929579607}a^{14}+\frac{7931581161872}{20827098594081}a^{13}+\frac{180469010776073}{229098084534891}a^{12}-\frac{1622996202995}{17622929579607}a^{11}-\frac{136128812607809}{76366028178297}a^{10}-\frac{90426556974634}{76366028178297}a^{9}-\frac{43561084298297}{20827098594081}a^{8}-\frac{16683930249527}{76366028178297}a^{7}+\frac{148624515425407}{229098084534891}a^{6}-\frac{51413143226327}{229098084534891}a^{5}+\frac{164815899999902}{229098084534891}a^{4}+\frac{39371319493906}{12057793922889}a^{3}-\frac{40957789892836}{25455342726099}a^{2}-\frac{8283175155095}{76366028178297}a+\frac{127417177388684}{229098084534891}$, $\frac{3492927407246}{229098084534891}a^{22}-\frac{24714257418748}{229098084534891}a^{21}+\frac{52054069855219}{229098084534891}a^{20}-\frac{8719362146452}{229098084534891}a^{19}-\frac{403756713715}{2633311316493}a^{18}-\frac{386422053650}{2314122066009}a^{17}+\frac{3246069455822}{20827098594081}a^{16}+\frac{20009495650006}{76366028178297}a^{15}+\frac{126276693537386}{229098084534891}a^{14}-\frac{71994420542974}{229098084534891}a^{13}+\frac{331519606602505}{229098084534891}a^{12}-\frac{502286841123514}{229098084534891}a^{11}-\frac{160713219513892}{76366028178297}a^{10}-\frac{55784546289476}{76366028178297}a^{9}-\frac{18845558486261}{17622929579607}a^{8}+\frac{156082970720366}{76366028178297}a^{7}+\frac{815661144127163}{229098084534891}a^{6}-\frac{411426910307659}{229098084534891}a^{5}+\frac{12\!\cdots\!43}{229098084534891}a^{4}+\frac{28660119560669}{12057793922889}a^{3}-\frac{140505148386104}{25455342726099}a^{2}-\frac{115230657357862}{76366028178297}a+\frac{16866341774366}{20827098594081}$, $\frac{1313169675203}{229098084534891}a^{22}-\frac{8139248918077}{229098084534891}a^{21}+\frac{10575481614319}{229098084534891}a^{20}+\frac{21676304168219}{229098084534891}a^{19}-\frac{14338501243031}{76366028178297}a^{18}+\frac{1739102417566}{25455342726099}a^{17}-\frac{31908647289323}{229098084534891}a^{16}+\frac{1711876726603}{5874309859869}a^{15}+\frac{23953845223277}{229098084534891}a^{14}+\frac{28317799275041}{229098084534891}a^{13}+\frac{103525976580427}{229098084534891}a^{12}-\frac{5652003586586}{20827098594081}a^{11}-\frac{12881174951738}{6942366198027}a^{10}+\frac{1367095597081}{5874309859869}a^{9}-\frac{368401137418946}{229098084534891}a^{8}+\frac{79075241577869}{76366028178297}a^{7}+\frac{287328269765726}{229098084534891}a^{6}-\frac{122121033437995}{229098084534891}a^{5}+\frac{32401671014195}{20827098594081}a^{4}+\frac{33150011248934}{12057793922889}a^{3}-\frac{67959406672849}{25455342726099}a^{2}-\frac{1493448837653}{6942366198027}a+\frac{205087405196056}{229098084534891}$, $\frac{99598199015}{12057793922889}a^{22}-\frac{672804892723}{12057793922889}a^{21}+\frac{1320144151186}{12057793922889}a^{20}-\frac{8128459349}{415785997341}a^{19}+\frac{22804525288}{4019264640963}a^{18}-\frac{28108902226}{148861653369}a^{17}+\frac{914142960916}{12057793922889}a^{16}+\frac{21136520791}{309174203151}a^{15}+\frac{492360611386}{1096163083899}a^{14}+\frac{1101884277170}{12057793922889}a^{13}+\frac{7276900392406}{12057793922889}a^{12}-\frac{10829878395418}{12057793922889}a^{11}-\frac{4946873771440}{4019264640963}a^{10}-\frac{3928196279963}{4019264640963}a^{9}-\frac{773022031115}{927522609453}a^{8}+\frac{1003118105255}{4019264640963}a^{7}+\frac{1452863145386}{927522609453}a^{6}-\frac{13273408853}{1096163083899}a^{5}+\frac{25359536509207}{12057793922889}a^{4}+\frac{31957274668688}{12057793922889}a^{3}-\frac{2994965295}{1711053487}a^{2}-\frac{1138222153132}{4019264640963}a+\frac{6734420182957}{12057793922889}$, $\frac{119166067783}{17622929579607}a^{22}-\frac{10056699525149}{229098084534891}a^{21}+\frac{18856103188601}{229098084534891}a^{20}-\frac{5043067010051}{229098084534891}a^{19}+\frac{3557366860964}{76366028178297}a^{18}-\frac{3352621503043}{25455342726099}a^{17}+\frac{7488596161733}{229098084534891}a^{16}-\frac{3760362620002}{76366028178297}a^{15}+\frac{54993202613179}{229098084534891}a^{14}+\frac{7567601036978}{20827098594081}a^{13}+\frac{15823772777678}{17622929579607}a^{12}-\frac{2126781796591}{7899933949479}a^{11}-\frac{3163494878068}{2633311316493}a^{10}-\frac{150641073426238}{76366028178297}a^{9}-\frac{526879685307784}{229098084534891}a^{8}-\frac{50429518724789}{76366028178297}a^{7}+\frac{11520460009358}{7899933949479}a^{6}+\frac{594152488967989}{229098084534891}a^{5}+\frac{770424694429118}{229098084534891}a^{4}+\frac{26699568960247}{12057793922889}a^{3}-\frac{27751229397326}{25455342726099}a^{2}-\frac{6821615865332}{5874309859869}a+\frac{206630383733819}{229098084534891}$, $\frac{968494162283}{229098084534891}a^{22}-\frac{413289565825}{17622929579607}a^{21}+\frac{369315564445}{17622929579607}a^{20}+\frac{14683824294803}{229098084534891}a^{19}-\frac{2480384693057}{76366028178297}a^{18}-\frac{585357393167}{8485114242033}a^{17}-\frac{13337928583280}{229098084534891}a^{16}+\frac{7708835554315}{76366028178297}a^{15}+\frac{26978159298830}{229098084534891}a^{14}+\frac{94682496981959}{229098084534891}a^{13}+\frac{63332589927487}{229098084534891}a^{12}+\frac{73586368761638}{229098084534891}a^{11}-\frac{104358004007872}{76366028178297}a^{10}-\frac{83780401299119}{76366028178297}a^{9}-\frac{430260796549097}{229098084534891}a^{8}-\frac{24332849594737}{76366028178297}a^{7}+\frac{7477238731741}{20827098594081}a^{6}+\frac{288470784516929}{229098084534891}a^{5}+\frac{279987029668486}{229098084534891}a^{4}+\frac{47774762408741}{12057793922889}a^{3}+\frac{26403745196}{217567031847}a^{2}-\frac{13324725876106}{76366028178297}a-\frac{100005355379621}{229098084534891}$, $\frac{287822909951}{25455342726099}a^{22}-\frac{1744354004504}{25455342726099}a^{21}+\frac{2460987159181}{25455342726099}a^{20}+\frac{683884865036}{8485114242033}a^{19}-\frac{598259283629}{25455342726099}a^{18}-\frac{6758097781069}{25455342726099}a^{17}+\frac{1564142785463}{25455342726099}a^{16}-\frac{126619515037}{2828371414011}a^{15}+\frac{1747019415308}{1958103286623}a^{14}+\frac{282098253124}{25455342726099}a^{13}+\frac{1506398392523}{877770438831}a^{12}-\frac{8647416292594}{8485114242033}a^{11}-\frac{46297318889339}{25455342726099}a^{10}-\frac{90976294913587}{25455342726099}a^{9}-\frac{41657143846792}{25455342726099}a^{8}-\frac{5359242939344}{2828371414011}a^{7}+\frac{9243700206538}{2314122066009}a^{6}-\frac{4706022857378}{25455342726099}a^{5}+\frac{8260895407936}{1958103286623}a^{4}+\frac{2332276818527}{446584960107}a^{3}+\frac{17744216242432}{25455342726099}a^{2}-\frac{18244977296149}{25455342726099}a+\frac{18689020830218}{25455342726099}$, $\frac{1918860519313}{229098084534891}a^{22}-\frac{14170261833128}{229098084534891}a^{21}+\frac{34992411001466}{229098084534891}a^{20}-\frac{32342466676211}{229098084534891}a^{19}+\frac{13414528392896}{76366028178297}a^{18}-\frac{933464101633}{2314122066009}a^{17}+\frac{86699269896119}{229098084534891}a^{16}-\frac{19153913040700}{76366028178297}a^{15}+\frac{15654183626693}{20827098594081}a^{14}-\frac{121580548320977}{229098084534891}a^{13}+\frac{297436432926362}{229098084534891}a^{12}-\frac{493287941850908}{229098084534891}a^{11}+\frac{41386169899378}{76366028178297}a^{10}-\frac{225780476220385}{76366028178297}a^{9}+\frac{30424030600414}{20827098594081}a^{8}-\frac{48256745597435}{76366028178297}a^{7}+\frac{668595445563280}{229098084534891}a^{6}-\frac{31361111971573}{20827098594081}a^{5}+\frac{966322694118464}{229098084534891}a^{4}-\frac{9431171065}{4548394539}a^{3}+\frac{64453661483141}{25455342726099}a^{2}-\frac{292555404171509}{76366028178297}a+\frac{327755091829655}{229098084534891}$, $\frac{27096204065}{2828371414011}a^{22}-\frac{63503061013}{942790471337}a^{21}+\frac{97527937775}{652701095541}a^{20}-\frac{743983597033}{8485114242033}a^{19}+\frac{574718748260}{8485114242033}a^{18}-\frac{149795620294}{652701095541}a^{17}+\frac{1598239545119}{8485114242033}a^{16}-\frac{82279933927}{942790471337}a^{15}+\frac{1491572328586}{2828371414011}a^{14}-\frac{469184520692}{2828371414011}a^{13}+\frac{10905904275755}{8485114242033}a^{12}-\frac{11626301006314}{8485114242033}a^{11}-\frac{3435324113440}{8485114242033}a^{10}-\frac{16812921614935}{8485114242033}a^{9}-\frac{9604377409291}{8485114242033}a^{8}-\frac{717993260828}{942790471337}a^{7}+\frac{1209834045960}{942790471337}a^{6}-\frac{1843822637515}{2828371414011}a^{5}+\frac{38707569988625}{8485114242033}a^{4}+\frac{84656318200}{40598632737}a^{3}+\frac{8050720356158}{8485114242033}a^{2}-\frac{58282461940}{652701095541}a+\frac{1163616914402}{8485114242033}$, $a$ 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:  \( 195904703.063 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 195904703.063 \cdot 1}{2\cdot\sqrt{14982213765271379362645677088782079}}\cr\approx \mathstrut & 0.964351631335 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
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 - 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);
 
[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 - 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);
 
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 - 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);
 
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 $23$ ${\href{/padicField/3.2.0.1}{2} }^{11}{,}\,{\href{/padicField/3.1.0.1}{1} }$ $23$ $23$ ${\href{/padicField/11.2.0.1}{2} }^{11}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.2.0.1}{2} }^{11}{,}\,{\href{/padicField/13.1.0.1}{1} }$ $23$ ${\href{/padicField/19.2.0.1}{2} }^{11}{,}\,{\href{/padicField/19.1.0.1}{1} }$ $23$ ${\href{/padicField/29.2.0.1}{2} }^{11}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }^{11}{,}\,{\href{/padicField/31.1.0.1}{1} }$ $23$ $23$ $23$ $23$ $23$ $23$

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
\(1279\) Copy content Toggle raw display $\Q_{1279}$$x$$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}$