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Properties

Label 23.1.58230945563...6903.1
Degree $23$
Signature $[1, 11]$
Discriminant $-\,1447^{11}$
Root discriminant $32.47$
Ramified prime $1447$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_{23}$ (as 23T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 99, -385, 569, -393, 240, -161, -130, 246, 75, -270, 265, -228, 25, 165, -139, 43, 17, -63, 68, -34, 16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 2*x^22 + 16*x^21 - 34*x^20 + 68*x^19 - 63*x^18 + 17*x^17 + 43*x^16 - 139*x^15 + 165*x^14 + 25*x^13 - 228*x^12 + 265*x^11 - 270*x^10 + 75*x^9 + 246*x^8 - 130*x^7 - 161*x^6 + 240*x^5 - 393*x^4 + 569*x^3 - 385*x^2 + 99*x + 1)
 
gp: K = bnfinit(x^23 - 2*x^22 + 16*x^21 - 34*x^20 + 68*x^19 - 63*x^18 + 17*x^17 + 43*x^16 - 139*x^15 + 165*x^14 + 25*x^13 - 228*x^12 + 265*x^11 - 270*x^10 + 75*x^9 + 246*x^8 - 130*x^7 - 161*x^6 + 240*x^5 - 393*x^4 + 569*x^3 - 385*x^2 + 99*x + 1, 1)
 

Normalized defining polynomial

\( x^{23} - 2 x^{22} + 16 x^{21} - 34 x^{20} + 68 x^{19} - 63 x^{18} + 17 x^{17} + 43 x^{16} - 139 x^{15} + 165 x^{14} + 25 x^{13} - 228 x^{12} + 265 x^{11} - 270 x^{10} + 75 x^{9} + 246 x^{8} - 130 x^{7} - 161 x^{6} + 240 x^{5} - 393 x^{4} + 569 x^{3} - 385 x^{2} + 99 x + 1 \)

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

Invariants

Degree:  $23$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-58230945563524325903440619509366903=-\,1447^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.47$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $1447$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{27} a^{15} - \frac{1}{27} a^{13} + \frac{4}{27} a^{12} - \frac{1}{9} a^{11} - \frac{4}{27} a^{10} + \frac{1}{27} a^{9} + \frac{1}{9} a^{8} + \frac{10}{27} a^{6} - \frac{7}{27} a^{5} - \frac{4}{9} a^{4} + \frac{4}{27} a^{3} + \frac{5}{27} a^{2} + \frac{1}{3} a - \frac{2}{27}$, $\frac{1}{27} a^{16} - \frac{1}{27} a^{14} + \frac{4}{27} a^{13} - \frac{1}{9} a^{12} - \frac{4}{27} a^{11} + \frac{1}{27} a^{10} + \frac{1}{9} a^{9} + \frac{1}{27} a^{7} + \frac{11}{27} a^{6} + \frac{2}{9} a^{5} - \frac{5}{27} a^{4} - \frac{4}{27} a^{3} - \frac{11}{27} a - \frac{1}{3}$, $\frac{1}{27} a^{17} + \frac{1}{27} a^{14} - \frac{1}{27} a^{13} + \frac{4}{27} a^{11} + \frac{2}{27} a^{10} + \frac{1}{27} a^{9} + \frac{1}{27} a^{8} - \frac{4}{27} a^{7} + \frac{4}{27} a^{6} - \frac{1}{9} a^{5} - \frac{4}{27} a^{4} + \frac{10}{27} a^{3} + \frac{1}{9} a^{2} + \frac{4}{9} a + \frac{13}{27}$, $\frac{1}{135} a^{18} - \frac{2}{135} a^{17} + \frac{1}{135} a^{15} - \frac{19}{135} a^{13} + \frac{13}{135} a^{12} + \frac{1}{9} a^{11} + \frac{4}{45} a^{10} + \frac{17}{135} a^{9} - \frac{4}{45} a^{8} + \frac{1}{15} a^{7} + \frac{64}{135} a^{6} - \frac{7}{135} a^{5} - \frac{7}{45} a^{4} + \frac{8}{27} a^{3} - \frac{13}{45} a^{2} + \frac{49}{135} a + \frac{4}{135}$, $\frac{1}{135} a^{19} + \frac{1}{135} a^{17} + \frac{1}{135} a^{16} + \frac{2}{135} a^{15} + \frac{1}{135} a^{14} - \frac{4}{135} a^{12} - \frac{13}{135} a^{11} - \frac{1}{15} a^{10} - \frac{2}{15} a^{9} + \frac{1}{27} a^{8} + \frac{2}{135} a^{7} + \frac{22}{45} a^{6} - \frac{1}{27} a^{5} - \frac{37}{135} a^{4} - \frac{29}{135} a^{3} - \frac{14}{135} a^{2} + \frac{4}{45} a - \frac{2}{135}$, $\frac{1}{405} a^{20} - \frac{1}{405} a^{19} - \frac{1}{135} a^{17} + \frac{1}{405} a^{16} - \frac{2}{405} a^{15} - \frac{7}{135} a^{14} - \frac{11}{81} a^{13} - \frac{67}{405} a^{12} - \frac{1}{405} a^{11} - \frac{16}{405} a^{10} - \frac{44}{405} a^{9} + \frac{34}{405} a^{8} + \frac{1}{9} a^{7} + \frac{20}{81} a^{6} + \frac{7}{81} a^{5} + \frac{199}{405} a^{4} + \frac{1}{81} a^{2} + \frac{1}{15} a + \frac{8}{405}$, $\frac{1}{2025} a^{21} + \frac{2}{2025} a^{19} - \frac{1}{405} a^{17} + \frac{17}{2025} a^{16} - \frac{14}{2025} a^{15} + \frac{92}{2025} a^{14} + \frac{241}{2025} a^{13} - \frac{86}{2025} a^{12} + \frac{244}{2025} a^{11} - \frac{8}{75} a^{10} + \frac{32}{2025} a^{9} + \frac{103}{2025} a^{8} + \frac{13}{2025} a^{7} - \frac{32}{135} a^{6} + \frac{62}{225} a^{5} - \frac{46}{405} a^{4} - \frac{112}{2025} a^{3} - \frac{937}{2025} a^{2} - \frac{127}{2025} a - \frac{346}{2025}$, $\frac{1}{104470934408325} a^{22} - \frac{914858467}{6964728960555} a^{21} - \frac{32762334668}{104470934408325} a^{20} - \frac{56589028498}{20894186881665} a^{19} - \frac{32608635196}{20894186881665} a^{18} + \frac{298230323552}{104470934408325} a^{17} - \frac{175588356248}{34823644802775} a^{16} - \frac{965948654048}{104470934408325} a^{15} + \frac{50294744476}{104470934408325} a^{14} + \frac{8343057920609}{104470934408325} a^{13} - \frac{15682282111}{104470934408325} a^{12} + \frac{1391427574309}{104470934408325} a^{11} - \frac{401488707002}{11607881600925} a^{10} - \frac{737935559353}{11607881600925} a^{9} + \frac{5414916967156}{34823644802775} a^{8} - \frac{150434843501}{2321576320185} a^{7} - \frac{13838276284492}{104470934408325} a^{6} + \frac{3788513561851}{20894186881665} a^{5} + \frac{38665881240733}{104470934408325} a^{4} - \frac{1968498721702}{104470934408325} a^{3} + \frac{7920828501611}{34823644802775} a^{2} - \frac{8471973936001}{104470934408325} a + \frac{6111746943359}{20894186881665}$

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

Class group and class number

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

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 530761705.173 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{23}$ (as 23T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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 $\Q$.

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $23$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $23$ $23$ $23$ $23$ $23$ $23$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $23$ $23$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $23$ $23$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
1447Data not computed