# 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)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining 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]);

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

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}$

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 $2$) 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