↑

Properties

Label 22.6.37909882314...9009.1
Degree $22$
Signature $[6, 8]$
Discriminant $23^{20}\cdot 47^{2}$
Root discriminant $24.54$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T23

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 10, -6, -363, -1234, -690, 1646, 1812, -164, -1131, -1069, 15, 156, 221, 79, 117, -63, -23, 3, -13, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 5*x^20 - 13*x^19 + 3*x^18 - 23*x^17 - 63*x^16 + 117*x^15 + 79*x^14 + 221*x^13 + 156*x^12 + 15*x^11 - 1069*x^10 - 1131*x^9 - 164*x^8 + 1812*x^7 + 1646*x^6 - 690*x^5 - 1234*x^4 - 363*x^3 - 6*x^2 + 10*x + 1)
 
gp: K = bnfinit(x^22 - x^21 + 5*x^20 - 13*x^19 + 3*x^18 - 23*x^17 - 63*x^16 + 117*x^15 + 79*x^14 + 221*x^13 + 156*x^12 + 15*x^11 - 1069*x^10 - 1131*x^9 - 164*x^8 + 1812*x^7 + 1646*x^6 - 690*x^5 - 1234*x^4 - 363*x^3 - 6*x^2 + 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 5 x^{20} - 13 x^{19} + 3 x^{18} - 23 x^{17} - 63 x^{16} + 117 x^{15} + 79 x^{14} + 221 x^{13} + 156 x^{12} + 15 x^{11} - 1069 x^{10} - 1131 x^{9} - 164 x^{8} + 1812 x^{7} + 1646 x^{6} - 690 x^{5} - 1234 x^{4} - 363 x^{3} - 6 x^{2} + 10 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3790988231418101231518884499009=23^{20}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.54$
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:  $23, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{10191388674737020354634217522157} a^{21} + \frac{2578164997823128219308452693194}{10191388674737020354634217522157} a^{20} + \frac{3646989572711473138287284176332}{10191388674737020354634217522157} a^{19} - \frac{2666282424612838352046043843336}{10191388674737020354634217522157} a^{18} + \frac{4612725007892737542519086806548}{10191388674737020354634217522157} a^{17} - \frac{359022680614952927966426151515}{10191388674737020354634217522157} a^{16} + \frac{994770508587392948082456499473}{10191388674737020354634217522157} a^{15} + \frac{362761085266824123180502526358}{10191388674737020354634217522157} a^{14} + \frac{422816268422604879419343950037}{10191388674737020354634217522157} a^{13} + \frac{3114620458084284577189703381608}{10191388674737020354634217522157} a^{12} - \frac{4288744378104634184082836077134}{10191388674737020354634217522157} a^{11} + \frac{634410985730960227553491493343}{10191388674737020354634217522157} a^{10} - \frac{3635261196064879603966701611220}{10191388674737020354634217522157} a^{9} + \frac{791713466399016619811614954709}{10191388674737020354634217522157} a^{8} + \frac{2798036149485325469688565855967}{10191388674737020354634217522157} a^{7} + \frac{4215178287154735736141911703023}{10191388674737020354634217522157} a^{6} - \frac{3438041228481728444889940901428}{10191388674737020354634217522157} a^{5} - \frac{3989835784734430046445045238474}{10191388674737020354634217522157} a^{4} - \frac{3216379188462000930694834995489}{10191388674737020354634217522157} a^{3} + \frac{2809904793246851608677841480297}{10191388674737020354634217522157} a^{2} - \frac{3402514124415836753151374698230}{10191388674737020354634217522157} a + \frac{529911411884370559284392578149}{10191388674737020354634217522157}$

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:  $13$
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:  \( 4141006.01086 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T23:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 11264
The 104 conjugacy class representatives for t22n23 are not computed
Character table for t22n23 is not computed

Intermediate fields

\(\Q(\zeta_{23})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 22 siblings: data not computed
Degree 44 siblings: 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
$23$23.11.10.10$x^{11} - 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
23.11.10.10$x^{11} - 23$$11$$1$$10$$C_{11}$$[\ ]_{11}$
47Data not computed