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Properties

Label 23.1.29398869933...0047.1
Degree $23$
Signature $[1, 11]$
Discriminant $-\,1103^{11}$
Root discriminant $28.52$
Ramified prime $1103$
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, 50, 244, 524, 89, 129, -28, 515, 165, 274, -16, -106, 257, -19, 140, -19, -34, 53, -14, 33, -17, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 + 5*x^21 - 17*x^20 + 33*x^19 - 14*x^18 + 53*x^17 - 34*x^16 - 19*x^15 + 140*x^14 - 19*x^13 + 257*x^12 - 106*x^11 - 16*x^10 + 274*x^9 + 165*x^8 + 515*x^7 - 28*x^6 + 129*x^5 + 89*x^4 + 524*x^3 + 244*x^2 + 50*x - 1)
 
gp: K = bnfinit(x^23 + 5*x^21 - 17*x^20 + 33*x^19 - 14*x^18 + 53*x^17 - 34*x^16 - 19*x^15 + 140*x^14 - 19*x^13 + 257*x^12 - 106*x^11 - 16*x^10 + 274*x^9 + 165*x^8 + 515*x^7 - 28*x^6 + 129*x^5 + 89*x^4 + 524*x^3 + 244*x^2 + 50*x - 1, 1)
 

Normalized defining polynomial

\( x^{23} + 5 x^{21} - 17 x^{20} + 33 x^{19} - 14 x^{18} + 53 x^{17} - 34 x^{16} - 19 x^{15} + 140 x^{14} - 19 x^{13} + 257 x^{12} - 106 x^{11} - 16 x^{10} + 274 x^{9} + 165 x^{8} + 515 x^{7} - 28 x^{6} + 129 x^{5} + 89 x^{4} + 524 x^{3} + 244 x^{2} + 50 x - 1 \)

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:  \(-2939886993377131174870795633320047=-\,1103^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.52$
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:  $1103$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{16} - \frac{2}{5} a^{15} + \frac{1}{5} a^{14} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{18} + \frac{2}{5} a^{16} - \frac{1}{5} a^{15} + \frac{1}{5} a^{14} - \frac{2}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{2}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5}$, $\frac{1}{35} a^{19} - \frac{3}{35} a^{18} + \frac{3}{35} a^{17} + \frac{1}{5} a^{16} - \frac{3}{35} a^{15} + \frac{11}{35} a^{14} + \frac{2}{35} a^{13} + \frac{2}{5} a^{12} - \frac{4}{35} a^{11} - \frac{1}{5} a^{10} - \frac{2}{35} a^{9} + \frac{4}{35} a^{8} - \frac{11}{35} a^{6} + \frac{1}{7} a^{5} - \frac{1}{5} a^{4} - \frac{2}{35} a^{3} + \frac{4}{35} a^{2} + \frac{1}{7} a - \frac{12}{35}$, $\frac{1}{665} a^{20} + \frac{3}{665} a^{19} - \frac{64}{665} a^{18} + \frac{18}{665} a^{17} - \frac{17}{665} a^{16} + \frac{43}{95} a^{15} + \frac{187}{665} a^{14} - \frac{331}{665} a^{13} + \frac{58}{133} a^{12} + \frac{137}{665} a^{11} + \frac{1}{35} a^{10} - \frac{239}{665} a^{9} + \frac{318}{665} a^{8} + \frac{276}{665} a^{7} + \frac{107}{665} a^{6} - \frac{243}{665} a^{5} + \frac{306}{665} a^{4} + \frac{258}{665} a^{3} - \frac{69}{665} a^{2} + \frac{12}{133} a + \frac{327}{665}$, $\frac{1}{23275} a^{21} + \frac{16}{23275} a^{20} - \frac{9}{3325} a^{19} + \frac{223}{3325} a^{18} - \frac{1759}{23275} a^{17} - \frac{452}{23275} a^{16} + \frac{48}{475} a^{15} + \frac{10061}{23275} a^{14} - \frac{10606}{23275} a^{13} - \frac{123}{4655} a^{12} + \frac{243}{931} a^{11} + \frac{8387}{23275} a^{10} - \frac{3644}{23275} a^{9} + \frac{6652}{23275} a^{8} - \frac{428}{23275} a^{7} + \frac{3694}{23275} a^{6} - \frac{3309}{23275} a^{5} - \frac{10793}{23275} a^{4} - \frac{363}{23275} a^{3} - \frac{5777}{23275} a^{2} - \frac{363}{3325} a - \frac{6199}{23275}$, $\frac{1}{3046699326689712481981375} a^{22} + \frac{25545655080006449148}{3046699326689712481981375} a^{21} + \frac{558005921733602553434}{3046699326689712481981375} a^{20} - \frac{9492408196367534356}{87048552191134642342325} a^{19} + \frac{252297766175310222540003}{3046699326689712481981375} a^{18} + \frac{13734561780472451840416}{609339865337942496396275} a^{17} + \frac{772252659968996159057218}{3046699326689712481981375} a^{16} - \frac{216995991261639503573164}{609339865337942496396275} a^{15} + \frac{324250576836940569126596}{3046699326689712481981375} a^{14} + \frac{265544620378028153801798}{3046699326689712481981375} a^{13} - \frac{32994940674651568428033}{609339865337942496396275} a^{12} - \frac{126669458345333881495338}{3046699326689712481981375} a^{11} - \frac{242586588547565610238551}{609339865337942496396275} a^{10} + \frac{316660290499975402807894}{3046699326689712481981375} a^{9} + \frac{853555952935271714185111}{3046699326689712481981375} a^{8} + \frac{1176735610661458585105118}{3046699326689712481981375} a^{7} + \frac{25529038672061223186983}{234361486668439421690875} a^{6} + \frac{154569711627625588223627}{435242760955673211711625} a^{5} - \frac{381941050928700821861474}{3046699326689712481981375} a^{4} - \frac{221239360814385883978063}{3046699326689712481981375} a^{3} + \frac{28477222848814287047257}{121867973067588499279255} a^{2} + \frac{1166011909557886455496869}{3046699326689712481981375} a + \frac{737938038390418032622912}{3046699326689712481981375}$

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:  \( 42306743.1121 \) (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 $23$ $23$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $23$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $23$ $23$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $23$ $23$ $23$ $23$ $23$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
1103Data not computed