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Properties

Label 23.1.15232494259...3439.1
Degree $23$
Signature $[1, 11]$
Discriminant $-\,1039^{11}$
Root discriminant $27.72$
Ramified prime $1039$
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, 49, 45, 28, 4, 76, -228, 40, 34, -17, 24, 19, 19, -62, 48, 31, -26, -14, -6, 25, -14, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 + 4*x^21 - 14*x^20 + 25*x^19 - 6*x^18 - 14*x^17 - 26*x^16 + 31*x^15 + 48*x^14 - 62*x^13 + 19*x^12 + 19*x^11 + 24*x^10 - 17*x^9 + 34*x^8 + 40*x^7 - 228*x^6 + 76*x^5 + 4*x^4 + 28*x^3 + 45*x^2 + 49*x + 1)
 
gp: K = bnfinit(x^23 + 4*x^21 - 14*x^20 + 25*x^19 - 6*x^18 - 14*x^17 - 26*x^16 + 31*x^15 + 48*x^14 - 62*x^13 + 19*x^12 + 19*x^11 + 24*x^10 - 17*x^9 + 34*x^8 + 40*x^7 - 228*x^6 + 76*x^5 + 4*x^4 + 28*x^3 + 45*x^2 + 49*x + 1, 1)
 

Normalized defining polynomial

\( x^{23} + 4 x^{21} - 14 x^{20} + 25 x^{19} - 6 x^{18} - 14 x^{17} - 26 x^{16} + 31 x^{15} + 48 x^{14} - 62 x^{13} + 19 x^{12} + 19 x^{11} + 24 x^{10} - 17 x^{9} + 34 x^{8} + 40 x^{7} - 228 x^{6} + 76 x^{5} + 4 x^{4} + 28 x^{3} + 45 x^{2} + 49 x + 1 \)

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:  \(-1523249425950439235855871175083439=-\,1039^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.72$
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:  $1039$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{19} + \frac{1}{27} a^{17} + \frac{1}{27} a^{16} - \frac{1}{9} a^{15} + \frac{1}{9} a^{14} - \frac{1}{9} a^{12} - \frac{2}{27} a^{11} - \frac{2}{27} a^{9} + \frac{1}{27} a^{8} + \frac{4}{9} a^{7} - \frac{4}{9} a^{6} + \frac{1}{3} a^{5} - \frac{2}{9} a^{4} + \frac{10}{27} a^{3} - \frac{1}{3} a^{2} + \frac{10}{27} a - \frac{11}{27}$, $\frac{1}{1647} a^{20} - \frac{1}{549} a^{19} + \frac{85}{1647} a^{18} - \frac{59}{1647} a^{17} + \frac{14}{549} a^{16} - \frac{65}{549} a^{15} - \frac{7}{61} a^{14} + \frac{41}{549} a^{13} - \frac{209}{1647} a^{12} + \frac{68}{549} a^{11} - \frac{107}{1647} a^{10} - \frac{257}{1647} a^{9} - \frac{14}{549} a^{8} - \frac{145}{549} a^{7} - \frac{83}{183} a^{6} - \frac{233}{549} a^{5} + \frac{28}{1647} a^{4} + \frac{83}{549} a^{3} + \frac{31}{1647} a^{2} - \frac{611}{1647} a + \frac{34}{183}$, $\frac{1}{1684881} a^{21} - \frac{175}{1684881} a^{20} - \frac{4151}{561627} a^{19} + \frac{719}{561627} a^{18} - \frac{36719}{1684881} a^{17} + \frac{56021}{1684881} a^{16} - \frac{16099}{187209} a^{15} + \frac{57115}{561627} a^{14} + \frac{205921}{1684881} a^{13} + \frac{124724}{1684881} a^{12} + \frac{22225}{561627} a^{11} + \frac{13586}{187209} a^{10} + \frac{61120}{1684881} a^{9} + \frac{129521}{1684881} a^{8} - \frac{71395}{187209} a^{7} + \frac{91517}{561627} a^{6} - \frac{308513}{1684881} a^{5} - \frac{731626}{1684881} a^{4} + \frac{161272}{561627} a^{3} - \frac{250190}{561627} a^{2} - \frac{388946}{1684881} a - \frac{435895}{1684881}$, $\frac{1}{27918564098931} a^{22} + \frac{784448}{9306188032977} a^{21} - \frac{2349680683}{27918564098931} a^{20} - \frac{23447370737}{3102062677659} a^{19} + \frac{1370586264214}{27918564098931} a^{18} + \frac{331475931283}{9306188032977} a^{17} - \frac{172682317417}{27918564098931} a^{16} - \frac{828516859133}{9306188032977} a^{15} - \frac{3899339361131}{27918564098931} a^{14} - \frac{699736799495}{9306188032977} a^{13} + \frac{5709962494}{34767825777} a^{12} + \frac{156855724351}{9306188032977} a^{11} + \frac{1688941533211}{27918564098931} a^{10} + \frac{75603573679}{3102062677659} a^{9} + \frac{3776769126722}{27918564098931} a^{8} - \frac{661738837027}{9306188032977} a^{7} - \frac{9968983470590}{27918564098931} a^{6} + \frac{1022530562671}{3102062677659} a^{5} + \frac{5743185482693}{27918564098931} a^{4} + \frac{476765817470}{9306188032977} a^{3} + \frac{6400961280025}{27918564098931} a^{2} + \frac{95511075958}{846017093907} a + \frac{3517882628618}{27918564098931}$

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:  \( 55921894.4803 \) (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 $23$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $23$ $23$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $23$ $23$ $23$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $23$ $23$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1039Data not computed