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Properties

Label 23.23.2812897349...1792.1
Degree $23$
Signature $[23, 0]$
Discriminant $2^{22}\cdot 13^{22}\cdot 23^{23}$
Root discriminant $519.02$
Ramified primes $2, 13, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $F_{23}$ (as 23T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-12722733426052, -41219689062851, 0, 69756396875594, 0, -34878198437797, 0, 8048815024107, 0, -1031899362065, 0, 80820089896, 0, -4064915764, 0, 134008212, 0, -2880267, 0, 38870, 0, -299, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 299*x^21 + 38870*x^19 - 2880267*x^17 + 134008212*x^15 - 4064915764*x^13 + 80820089896*x^11 - 1031899362065*x^9 + 8048815024107*x^7 - 34878198437797*x^5 + 69756396875594*x^3 - 41219689062851*x - 12722733426052)
 
gp: K = bnfinit(x^23 - 299*x^21 + 38870*x^19 - 2880267*x^17 + 134008212*x^15 - 4064915764*x^13 + 80820089896*x^11 - 1031899362065*x^9 + 8048815024107*x^7 - 34878198437797*x^5 + 69756396875594*x^3 - 41219689062851*x - 12722733426052, 1)
 

Normalized defining polynomial

\( x^{23} - 299 x^{21} + 38870 x^{19} - 2880267 x^{17} + 134008212 x^{15} - 4064915764 x^{13} + 80820089896 x^{11} - 1031899362065 x^{9} + 8048815024107 x^{7} - 34878198437797 x^{5} + 69756396875594 x^{3} - 41219689062851 x - 12722733426052 \)

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

Invariants

Degree:  $23$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[23, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(281289734904574222488347652007286105552626963997840714384801792=2^{22}\cdot 13^{22}\cdot 23^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $519.02$
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:  $2, 13, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{87263423403} a^{12} + \frac{2541706348}{6712571031} a^{11} - \frac{4}{2237523677} a^{10} - \frac{985172090}{6712571031} a^{9} + \frac{234}{2237523677} a^{8} - \frac{2471619568}{6712571031} a^{7} - \frac{18928}{6712571031} a^{6} + \frac{2528776924}{6712571031} a^{5} + \frac{76895}{2237523677} a^{4} + \frac{1450906678}{6712571031} a^{3} - \frac{342732}{2237523677} a^{2} + \frac{1597699462}{6712571031} a + \frac{742586}{6712571031}$, $\frac{1}{87263423403} a^{13} - \frac{13}{6712571031} a^{11} - \frac{520672631}{6712571031} a^{10} + \frac{845}{6712571031} a^{9} + \frac{561731720}{6712571031} a^{8} - \frac{8788}{2237523677} a^{7} + \frac{1291490864}{6712571031} a^{6} + \frac{399854}{6712571031} a^{5} - \frac{1929239801}{6712571031} a^{4} - \frac{2599051}{6712571031} a^{3} + \frac{2471202160}{6712571031} a^{2} + \frac{4826809}{6712571031} a + \frac{114978166}{6712571031}$, $\frac{1}{87263423403} a^{14} - \frac{576845803}{6712571031} a^{11} - \frac{1183}{6712571031} a^{10} + \frac{627308095}{2237523677} a^{9} + \frac{30758}{2237523677} a^{8} - \frac{232812206}{6712571031} a^{7} - \frac{2798978}{6712571031} a^{6} + \frac{2542085402}{6712571031} a^{5} + \frac{36386714}{6712571031} a^{4} - \frac{690697405}{6712571031} a^{3} - \frac{168938315}{6712571031} a^{2} + \frac{1623346004}{6712571031} a + \frac{125497034}{6712571031}$, $\frac{1}{87263423403} a^{15} - \frac{455}{2237523677} a^{11} - \frac{280865860}{2237523677} a^{10} + \frac{118300}{6712571031} a^{9} + \frac{1406297668}{6712571031} a^{8} - \frac{1384110}{2237523677} a^{7} - \frac{1129133495}{6712571031} a^{6} + \frac{22391824}{2237523677} a^{5} + \frac{1679178269}{6712571031} a^{4} - \frac{454833925}{6712571031} a^{3} - \frac{500867704}{6712571031} a^{2} + \frac{289608540}{2237523677} a + \frac{788313119}{6712571031}$, $\frac{1}{87263423403} a^{16} - \frac{9403203}{2237523677} a^{11} - \frac{94640}{6712571031} a^{10} - \frac{937474658}{6712571031} a^{9} + \frac{2768220}{2237523677} a^{8} - \frac{79251101}{6712571031} a^{7} - \frac{89567296}{2237523677} a^{6} + \frac{1288352414}{6712571031} a^{5} - \frac{3073899631}{6712571031} a^{4} + \frac{3128229521}{6712571031} a^{3} + \frac{920400231}{2237523677} a^{2} - \frac{1934768635}{6712571031} a - \frac{82651164}{2237523677}$, $\frac{1}{87263423403} a^{17} - \frac{114920}{6712571031} a^{11} + \frac{1374397369}{6712571031} a^{10} + \frac{3734900}{2237523677} a^{9} + \frac{2283941455}{6712571031} a^{8} - \frac{139834656}{2237523677} a^{7} + \frac{717569492}{6712571031} a^{6} + \frac{356847689}{6712571031} a^{5} + \frac{2533003973}{6712571031} a^{4} - \frac{748484861}{2237523677} a^{3} - \frac{1775004004}{6712571031} a^{2} + \frac{676412192}{2237523677} a + \frac{831686241}{2237523677}$, $\frac{1}{87263423403} a^{18} + \frac{511271061}{2237523677} a^{11} - \frac{2240940}{2237523677} a^{10} - \frac{124935618}{2237523677} a^{9} + \frac{209751984}{2237523677} a^{8} - \frac{663836697}{2237523677} a^{7} - \frac{356847689}{2237523677} a^{6} - \frac{1283003066}{6712571031} a^{5} + \frac{15861812}{2237523677} a^{4} - \frac{1821385520}{6712571031} a^{3} + \frac{1041435505}{2237523677} a^{2} - \frac{1411891954}{6712571031} a + \frac{1819560445}{6712571031}$, $\frac{1}{87263423403} a^{19} - \frac{2838524}{2237523677} a^{11} - \frac{917502474}{2237523677} a^{10} + \frac{295206496}{2237523677} a^{9} + \frac{950850839}{2237523677} a^{8} - \frac{325434959}{2237523677} a^{7} + \frac{617840671}{6712571031} a^{6} + \frac{419984043}{2237523677} a^{5} + \frac{603758491}{6712571031} a^{4} - \frac{153958152}{2237523677} a^{3} + \frac{3313701983}{6712571031} a^{2} + \frac{403517320}{6712571031} a - \frac{764722372}{2237523677}$, $\frac{1}{87263423403} a^{20} - \frac{112465016}{2237523677} a^{11} - \frac{147603248}{2237523677} a^{10} - \frac{55798128}{2237523677} a^{9} + \frac{966174618}{2237523677} a^{8} + \frac{3265945810}{6712571031} a^{7} + \frac{68801731}{2237523677} a^{6} - \frac{905636900}{6712571031} a^{5} + \frac{769790760}{2237523677} a^{4} - \frac{2808196765}{6712571031} a^{3} + \frac{1668604531}{6712571031} a^{2} + \frac{513380591}{2237523677} a - \frac{926092387}{2237523677}$, $\frac{1}{87263423403} a^{21} - \frac{193729263}{2237523677} a^{11} + \frac{299848792}{2237523677} a^{10} + \frac{849623732}{2237523677} a^{9} + \frac{1263050629}{6712571031} a^{8} - \frac{583123228}{2237523677} a^{7} - \frac{602216564}{6712571031} a^{6} - \frac{620963081}{2237523677} a^{5} + \frac{206438921}{6712571031} a^{4} + \frac{3012422929}{6712571031} a^{3} + \frac{573469888}{2237523677} a^{2} - \frac{300716057}{2237523677} a + \frac{826750271}{2237523677}$, $\frac{1}{87263423403} a^{22} + \frac{424325214}{2237523677} a^{11} - \frac{284333495}{2237523677} a^{10} + \frac{1563233839}{6712571031} a^{9} - \frac{253573920}{2237523677} a^{8} - \frac{2786601422}{6712571031} a^{7} + \frac{23604572}{2237523677} a^{6} - \frac{443088292}{6712571031} a^{5} - \frac{2514547238}{6712571031} a^{4} + \frac{26741345}{2237523677} a^{3} + \frac{1070352950}{2237523677} a^{2} - \frac{884288665}{2237523677} a - \frac{1114523376}{2237523677}$

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:  $22$
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:  \( 3930853533530000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_{23}$ (as 23T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 506
The 23 conjugacy class representatives for $F_{23}$
Character table for $F_{23}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 46 sibling: 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 R $22{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $22{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R $22{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ $22{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $22{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $22{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $22{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. 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
2Data not computed
13Data not computed
23Data not computed