Properties

Label 23.23.824...241.1
Degree $23$
Signature $[23, 0]$
Discriminant $8.242\times 10^{59}$
Root discriminant $402.76$
Ramified prime $23$
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 - 253*x^21 - 161*x^20 + 26335*x^19 + 33028*x^18 - 1467515*x^17 - 2734079*x^16 + 47658093*x^15 + 118763283*x^14 - 910024555*x^13 - 2931964974*x^12 + 9516128606*x^11 + 41347298260*x^10 - 39738537224*x^9 - 313669878607*x^8 - 104709853475*x^7 + 1076728692839*x^6 + 1230914947669*x^5 - 913297744524*x^4 - 1823221987393*x^3 - 124828056170*x^2 + 645413252986*x + 157112485811)
 
gp: K = bnfinit(x^23 - 253*x^21 - 161*x^20 + 26335*x^19 + 33028*x^18 - 1467515*x^17 - 2734079*x^16 + 47658093*x^15 + 118763283*x^14 - 910024555*x^13 - 2931964974*x^12 + 9516128606*x^11 + 41347298260*x^10 - 39738537224*x^9 - 313669878607*x^8 - 104709853475*x^7 + 1076728692839*x^6 + 1230914947669*x^5 - 913297744524*x^4 - 1823221987393*x^3 - 124828056170*x^2 + 645413252986*x + 157112485811, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![157112485811, 645413252986, -124828056170, -1823221987393, -913297744524, 1230914947669, 1076728692839, -104709853475, -313669878607, -39738537224, 41347298260, 9516128606, -2931964974, -910024555, 118763283, 47658093, -2734079, -1467515, 33028, 26335, -161, -253, 0, 1]);
 

\(x^{23} - 253 x^{21} - 161 x^{20} + 26335 x^{19} + 33028 x^{18} - 1467515 x^{17} - 2734079 x^{16} + 47658093 x^{15} + 118763283 x^{14} - 910024555 x^{13} - 2931964974 x^{12} + 9516128606 x^{11} + 41347298260 x^{10} - 39738537224 x^{9} - 313669878607 x^{8} - 104709853475 x^{7} + 1076728692839 x^{6} + 1230914947669 x^{5} - 913297744524 x^{4} - 1823221987393 x^{3} - 124828056170 x^{2} + 645413252986 x + 157112485811\)  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:  \(824185149135487077883465900577270766354751717380230010246241\)\(\medspace = 23^{44}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $402.76$
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:  $23$
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:  \(529=23^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{529}(1,·)$, $\chi_{529}(323,·)$, $\chi_{529}(70,·)$, $\chi_{529}(392,·)$, $\chi_{529}(139,·)$, $\chi_{529}(461,·)$, $\chi_{529}(208,·)$, $\chi_{529}(277,·)$, $\chi_{529}(24,·)$, $\chi_{529}(346,·)$, $\chi_{529}(93,·)$, $\chi_{529}(415,·)$, $\chi_{529}(162,·)$, $\chi_{529}(484,·)$, $\chi_{529}(231,·)$, $\chi_{529}(300,·)$, $\chi_{529}(47,·)$, $\chi_{529}(369,·)$, $\chi_{529}(116,·)$, $\chi_{529}(438,·)$, $\chi_{529}(185,·)$, $\chi_{529}(507,·)$, $\chi_{529}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{22} - \frac{91367638590606636144038924400311825045394073983514780267800405437744889354351817030966712700380613823324595817745185764}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{21} - \frac{51246568935882907356354911948907447264068090767502429658912844484895460827567048943460642470908496493432718956495794650}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{20} + \frac{52720199530786192158376783883817634983680875693861951616763807433978898277298586063297138617998721418715209522770529219}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{19} - \frac{124048790256397822648293868846888130142420091290367348545869724334436188600400951252078688213247152312501960351346834085}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{18} - \frac{53051193772767301397002577726978772659121098821281711341215399896167021739823453056435056296656773667239389127645107835}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{17} + \frac{62417029703294823484348290524918360961164156881084160102151400762259944060530839727229447860250318531451073737821780265}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{16} - \frac{113945467350577699543630187525212271633049039093993479802083814252808046550156666239028275875296803341068444613506048455}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{15} - \frac{77828590013367730011933610869582897575056927481289294046244805427325710221419437127432227185817262511576077578137588}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{14} - \frac{84706665462422776941366299635439821914438448315492074158564244986429906059223445714971017304389482536662946180054995504}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{13} - \frac{56532716992193196343901527785886208950318000601664217651108107567860607439133240383516736669313979896714100526621148379}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{12} - \frac{120726228264909561649977979428001751873666378572949418287658601143910081116173873736359604233021165937225186198763725359}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{11} + \frac{62092376756626763723960709753670479390450664102252033413125257736824856744429285120944417200963021742514867525464775577}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{10} + \frac{22709917676881708499745811526632198790302449586538265753107991804593778235857588387356052218548604788870124088770304097}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{9} - \frac{17908528539699672442798922801522249875121280369108618915174210304926028240095639325916457213717044795000614123600533245}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{8} - \frac{20185926541663170708263237543003691244001290070425026832777518963358982885432612937521819582660326922351621693524415355}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{7} + \frac{70406286332429546966724350207237437256673260429864701965949512491735050820277129520112069720321856393299035396049777517}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{6} - \frac{25447361860961936534201284936789114835093712193868846197842662415735518874702994536839460887781198370488491148781567873}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{5} + \frac{87412376728682330922524535078333930327234632702354614783771672515977260053951385928805741525031838454911621012193305731}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{4} + \frac{85487346173535466326420486638448636266039313007474436531091733926636000353501968372179181216244534442558253352284595191}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{3} - \frac{61339521335177929844685979039363916732598544828034818464218232642510991384377933252776356640902044244278663049255077207}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a^{2} + \frac{17680146624897341443909378920451566727538727840963489359741781302469600787159094516721754418379321555187214697453076082}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449} a + \frac{83730608330391291332006649153079395983833022537278156764707533741794168031673506135317520747977788395015931537287936837}{249966391231401994087712687297404400445849260138999633581143624941110121511590682070695527021517583004662785208425669449}$  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:  \( 26795387302400000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{23}\cdot(2\pi)^{0}\cdot 26795387302400000000000 \cdot 1}{2\sqrt{824185149135487077883465900577270766354751717380230010246241}}\approx 0.123796267561194$ (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 $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ R $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$

In the table, R denotes a ramified prime. 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
23Data not computed