Properties

Label 23.23.399...721.1
Degree $23$
Signature $[23, 0]$
Discriminant $3.994\times 10^{58}$
Root discriminant $353.09$
Ramified prime $461$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{23}$ (as 23T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^23 - x^22 - 220*x^21 + 87*x^20 + 19764*x^19 + 7502*x^18 - 942100*x^17 - 1247222*x^16 + 25503060*x^15 + 62076437*x^14 - 368245290*x^13 - 1464919087*x^12 + 1795768610*x^11 + 16428532486*x^10 + 16509241674*x^9 - 60887740509*x^8 - 181108379413*x^7 - 148388500129*x^6 + 88587002490*x^5 + 252590462192*x^4 + 171337144724*x^3 + 31220491402*x^2 - 10221385976*x - 3012210881)
 
gp: K = bnfinit(x^23 - x^22 - 220*x^21 + 87*x^20 + 19764*x^19 + 7502*x^18 - 942100*x^17 - 1247222*x^16 + 25503060*x^15 + 62076437*x^14 - 368245290*x^13 - 1464919087*x^12 + 1795768610*x^11 + 16428532486*x^10 + 16509241674*x^9 - 60887740509*x^8 - 181108379413*x^7 - 148388500129*x^6 + 88587002490*x^5 + 252590462192*x^4 + 171337144724*x^3 + 31220491402*x^2 - 10221385976*x - 3012210881, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3012210881, -10221385976, 31220491402, 171337144724, 252590462192, 88587002490, -148388500129, -181108379413, -60887740509, 16509241674, 16428532486, 1795768610, -1464919087, -368245290, 62076437, 25503060, -1247222, -942100, 7502, 19764, 87, -220, -1, 1]);
 

\(x^{23} - x^{22} - 220 x^{21} + 87 x^{20} + 19764 x^{19} + 7502 x^{18} - 942100 x^{17} - 1247222 x^{16} + 25503060 x^{15} + 62076437 x^{14} - 368245290 x^{13} - 1464919087 x^{12} + 1795768610 x^{11} + 16428532486 x^{10} + 16509241674 x^{9} - 60887740509 x^{8} - 181108379413 x^{7} - 148388500129 x^{6} + 88587002490 x^{5} + 252590462192 x^{4} + 171337144724 x^{3} + 31220491402 x^{2} - 10221385976 x - 3012210881\)  Toggle raw display

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

Invariants

Degree:  $23$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[23, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(39941131008096550428701851231676209894223228004485139129721\)\(\medspace = 461^{22}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $353.09$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $461$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $23$
This field is Galois and abelian over $\Q$.
Conductor:  \(461\)
Dirichlet character group:    $\lbrace$$\chi_{461}(1,·)$, $\chi_{461}(322,·)$, $\chi_{461}(196,·)$, $\chi_{461}(262,·)$, $\chi_{461}(292,·)$, $\chi_{461}(14,·)$, $\chi_{461}(400,·)$, $\chi_{461}(23,·)$, $\chi_{461}(68,·)$, $\chi_{461}(348,·)$, $\chi_{461}(30,·)$, $\chi_{461}(416,·)$, $\chi_{461}(33,·)$, $\chi_{461}(420,·)$, $\chi_{461}(229,·)$, $\chi_{461}(167,·)$, $\chi_{461}(298,·)$, $\chi_{461}(359,·)$, $\chi_{461}(181,·)$, $\chi_{461}(439,·)$, $\chi_{461}(440,·)$, $\chi_{461}(441,·)$, $\chi_{461}(153,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{347} a^{21} + \frac{173}{347} a^{20} - \frac{98}{347} a^{19} + \frac{107}{347} a^{18} - \frac{144}{347} a^{17} - \frac{49}{347} a^{16} - \frac{100}{347} a^{15} + \frac{15}{347} a^{14} + \frac{49}{347} a^{13} + \frac{82}{347} a^{12} + \frac{4}{347} a^{11} - \frac{154}{347} a^{10} - \frac{47}{347} a^{9} - \frac{104}{347} a^{8} - \frac{40}{347} a^{7} + \frac{167}{347} a^{6} + \frac{67}{347} a^{5} - \frac{11}{347} a^{4} + \frac{49}{347} a^{3} - \frac{15}{347} a^{2} - \frac{115}{347} a$, $\frac{1}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{22} - \frac{698207311068754356132058721093456193119615987896004367573240188097806935588444588639909440176177}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{21} + \frac{358270334149022715447634482874439805055485214839257229207367778931503368648109405786377034188212574}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{20} + \frac{221598037027128510909850346479511962820212385873848134750782579656891178988033023794663067983664622}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{19} - \frac{370163384272409444263548922394425149840571066875260496508467940450069396761553522093933686041652033}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{18} + \frac{354998821282110700922994683602097618883478426036235050494410155607652506801716264571552407496699497}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{17} - \frac{30853291631755867121053968702779003111330281485927515853855112024449722978258545849930294298256284}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{16} - \frac{211605859336090071093890397231421727068726080824119335673111864126691888195423897710878935817941044}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{15} + \frac{315298847393384095103830821782714065448680835392376302460579444912640024481544241782537604823973880}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{14} + \frac{60633700590527972098735835240730866782309593710624489177720920893550495611942118136103463850207385}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{13} - \frac{182478179842682133504405291158951142711208756913820185561179258823843526062819175284437994886688131}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{12} - \frac{440337404088736104095558875471995004523078680470077566786137016021238383136397302059417440296480183}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{11} - \frac{365646142789714127271614280323185828657431510452462865123237268797244614399104645198824862870375616}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{10} - \frac{294237939104761202256800239016152693717743826036858147816353431910997647243415912373553912845357533}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{9} - \frac{246897761915466300094622757330679535808940979199107947487499962337240697210910857567490234400169907}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{8} + \frac{160411442535133355758160500067974842460249060835772950762684144056142721457903762542270613553082774}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{7} - \frac{156005517562328268735759836217930469643630047434073455393320842610109371161874434288194108554530640}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{6} - \frac{381133488358997592011226987997314516195500984255481523826702558594158985722888859963650303980741153}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{5} + \frac{138278096758643531922506304329010015671409873340176053867635785279712649081479818357225505243218057}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{4} + \frac{259178229215160251070117203830225292618835436270453459186168058807289245846384985796645695861516949}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{3} + \frac{440824067025862853823447131006299008607425627124502298281148452219723996739344195461539449317450625}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a^{2} - \frac{169187442196761164624713137158193221114538248590768579771516824867060863793520473943199429651586303}{888240951837481757759049950490502194483428549272408564095621684681177376743954250513895591541514679} a - \frac{833593163811045124096867139281328571697668634206701028895618248688548294717666551928976795911494}{2559772195497065584320028675765136007156854608854203354742425604268522699550300433757624183116757}$  Toggle raw display

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $22$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 5504325396653325000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{23}\cdot(2\pi)^{0}\cdot 5504325396653325000000 \cdot 1}{2\sqrt{39941131008096550428701851231676209894223228004485139129721}}\approx 0.115519107609513$ (assuming GRH)

Galois group

$C_{23}$ (as 23T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 23
The 23 conjugacy class representatives for $C_{23}$
Character table for $C_{23}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
461Data not computed