Properties

Label 23.23.140...521.1
Degree $23$
Signature $[23, 0]$
Discriminant $1.401\times 10^{47}$
Root discriminant $112.16$
Ramified prime $139$
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 - 66*x^21 + 147*x^20 + 1630*x^19 - 5648*x^18 - 16457*x^17 + 92686*x^16 + 18441*x^15 - 709360*x^14 + 832638*x^13 + 2239299*x^12 - 5679764*x^11 - 156443*x^10 + 12673530*x^9 - 11318727*x^8 - 6468097*x^7 + 14166332*x^6 - 3186420*x^5 - 5386949*x^4 + 2918745*x^3 + 436718*x^2 - 516219*x + 63941)
 
gp: K = bnfinit(x^23 - x^22 - 66*x^21 + 147*x^20 + 1630*x^19 - 5648*x^18 - 16457*x^17 + 92686*x^16 + 18441*x^15 - 709360*x^14 + 832638*x^13 + 2239299*x^12 - 5679764*x^11 - 156443*x^10 + 12673530*x^9 - 11318727*x^8 - 6468097*x^7 + 14166332*x^6 - 3186420*x^5 - 5386949*x^4 + 2918745*x^3 + 436718*x^2 - 516219*x + 63941, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![63941, -516219, 436718, 2918745, -5386949, -3186420, 14166332, -6468097, -11318727, 12673530, -156443, -5679764, 2239299, 832638, -709360, 18441, 92686, -16457, -5648, 1630, 147, -66, -1, 1]);
 

\(x^{23} - x^{22} - 66 x^{21} + 147 x^{20} + 1630 x^{19} - 5648 x^{18} - 16457 x^{17} + 92686 x^{16} + 18441 x^{15} - 709360 x^{14} + 832638 x^{13} + 2239299 x^{12} - 5679764 x^{11} - 156443 x^{10} + 12673530 x^{9} - 11318727 x^{8} - 6468097 x^{7} + 14166332 x^{6} - 3186420 x^{5} - 5386949 x^{4} + 2918745 x^{3} + 436718 x^{2} - 516219 x + 63941\)  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:  \(140063703503689367173618364344202364099995564521\)\(\medspace = 139^{22}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $112.16$
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:  $139$
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:  \(139\)
Dirichlet character group:    $\lbrace$$\chi_{139}(64,·)$, $\chi_{139}(1,·)$, $\chi_{139}(131,·)$, $\chi_{139}(6,·)$, $\chi_{139}(129,·)$, $\chi_{139}(77,·)$, $\chi_{139}(79,·)$, $\chi_{139}(80,·)$, $\chi_{139}(100,·)$, $\chi_{139}(91,·)$, $\chi_{139}(116,·)$, $\chi_{139}(34,·)$, $\chi_{139}(36,·)$, $\chi_{139}(65,·)$, $\chi_{139}(106,·)$, $\chi_{139}(44,·)$, $\chi_{139}(45,·)$, $\chi_{139}(112,·)$, $\chi_{139}(52,·)$, $\chi_{139}(55,·)$, $\chi_{139}(57,·)$, $\chi_{139}(125,·)$, $\chi_{139}(63,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{43} a^{19} - \frac{12}{43} a^{18} + \frac{19}{43} a^{17} - \frac{9}{43} a^{16} + \frac{11}{43} a^{14} - \frac{20}{43} a^{13} + \frac{18}{43} a^{10} + \frac{2}{43} a^{8} + \frac{2}{43} a^{7} + \frac{11}{43} a^{6} + \frac{21}{43} a^{5} - \frac{20}{43} a^{4} + \frac{21}{43} a^{3} - \frac{2}{43} a$, $\frac{1}{43} a^{20} + \frac{4}{43} a^{18} + \frac{4}{43} a^{17} + \frac{21}{43} a^{16} + \frac{11}{43} a^{15} - \frac{17}{43} a^{14} + \frac{18}{43} a^{13} + \frac{18}{43} a^{11} + \frac{1}{43} a^{10} + \frac{2}{43} a^{9} - \frac{17}{43} a^{8} - \frac{8}{43} a^{7} - \frac{19}{43} a^{6} + \frac{17}{43} a^{5} - \frac{4}{43} a^{4} - \frac{6}{43} a^{3} - \frac{2}{43} a^{2} + \frac{19}{43} a$, $\frac{1}{754951} a^{21} - \frac{5533}{754951} a^{20} + \frac{7720}{754951} a^{19} + \frac{366192}{754951} a^{18} + \frac{363702}{754951} a^{17} + \frac{321430}{754951} a^{16} - \frac{74081}{754951} a^{15} - \frac{87043}{754951} a^{14} - \frac{298075}{754951} a^{13} + \frac{72645}{754951} a^{12} - \frac{85747}{754951} a^{11} - \frac{242076}{754951} a^{10} - \frac{312169}{754951} a^{9} + \frac{332913}{754951} a^{8} + \frac{288609}{754951} a^{7} + \frac{242824}{754951} a^{6} + \frac{361274}{754951} a^{5} + \frac{1930}{7783} a^{4} - \frac{318747}{754951} a^{3} + \frac{161757}{754951} a^{2} - \frac{52318}{754951} a + \frac{7827}{17557}$, $\frac{1}{1424880116117105488551658527949} a^{22} + \frac{760445898851806163691671}{1424880116117105488551658527949} a^{21} - \frac{11731147140587931675190947439}{1424880116117105488551658527949} a^{20} + \frac{16517348045468643320241954208}{1424880116117105488551658527949} a^{19} - \frac{392266998589102181978954831679}{1424880116117105488551658527949} a^{18} - \frac{453160600122698737131456859472}{1424880116117105488551658527949} a^{17} - \frac{591812284922908894325999876630}{1424880116117105488551658527949} a^{16} - \frac{304118660702128975433196696289}{1424880116117105488551658527949} a^{15} + \frac{591299700161782929479439974537}{1424880116117105488551658527949} a^{14} - \frac{224213291241531700816956649405}{1424880116117105488551658527949} a^{13} - \frac{703611615285089579928594527900}{1424880116117105488551658527949} a^{12} + \frac{652026345437595812768557348917}{1424880116117105488551658527949} a^{11} - \frac{4471401494433208405902384462}{14689485733166035964450087917} a^{10} - \frac{115845137743880685202933417833}{1424880116117105488551658527949} a^{9} - \frac{142357881441246291245340579092}{1424880116117105488551658527949} a^{8} + \frac{270211501011554981974562726251}{1424880116117105488551658527949} a^{7} - \frac{555497250160671734426512263033}{1424880116117105488551658527949} a^{6} - \frac{79940787894248029944425140770}{1424880116117105488551658527949} a^{5} + \frac{173163381751297406804763935603}{1424880116117105488551658527949} a^{4} - \frac{278515365747601056858491545903}{1424880116117105488551658527949} a^{3} + \frac{602682006965624574094796952891}{1424880116117105488551658527949} a^{2} - \frac{239609006807047678926723835309}{1424880116117105488551658527949} a + \frac{1212150684671470448218646809}{33136746886444313687247872743}$  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:  \( 22066241000437346.10236048207 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{23}\cdot(2\pi)^{0}\cdot 22066241000437346.10236048207 \cdot 1}{2\sqrt{140063703503689367173618364344202364099995564521}}\approx 0.247300764485865$ (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 $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ $23$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{23}$ $23$ $23$ $23$

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
139Data not computed