↑

Properties

Label 22.12.1781764468...3423.2
Degree $22$
Signature $[12, 5]$
Discriminant $-\,23^{20}\cdot 47^{3}$
Root discriminant $29.24$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T28

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -1081, 4687, -3524, -12946, 25694, -3288, -29522, 22070, 9906, -16878, 969, 6548, -1820, -1739, 915, 219, -251, 32, 23, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 + 23*x^20 + 32*x^19 - 251*x^18 + 219*x^17 + 915*x^16 - 1739*x^15 - 1820*x^14 + 6548*x^13 + 969*x^12 - 16878*x^11 + 9906*x^10 + 22070*x^9 - 29522*x^8 - 3288*x^7 + 25694*x^6 - 12946*x^5 - 3524*x^4 + 4687*x^3 - 1081*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^22 - 9*x^21 + 23*x^20 + 32*x^19 - 251*x^18 + 219*x^17 + 915*x^16 - 1739*x^15 - 1820*x^14 + 6548*x^13 + 969*x^12 - 16878*x^11 + 9906*x^10 + 22070*x^9 - 29522*x^8 - 3288*x^7 + 25694*x^6 - 12946*x^5 - 3524*x^4 + 4687*x^3 - 1081*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{21} + 23 x^{20} + 32 x^{19} - 251 x^{18} + 219 x^{17} + 915 x^{16} - 1739 x^{15} - 1820 x^{14} + 6548 x^{13} + 969 x^{12} - 16878 x^{11} + 9906 x^{10} + 22070 x^{9} - 29522 x^{8} - 3288 x^{7} + 25694 x^{6} - 12946 x^{5} - 3524 x^{4} + 4687 x^{3} - 1081 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-178176446876650757881387571453423=-\,23^{20}\cdot 47^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{154786433280020486873} a^{21} - \frac{64849291404396475701}{154786433280020486873} a^{20} + \frac{76304523266753349259}{154786433280020486873} a^{19} + \frac{49913474583088740990}{154786433280020486873} a^{18} - \frac{20149473244998771679}{154786433280020486873} a^{17} + \frac{48440821994038416319}{154786433280020486873} a^{16} - \frac{51797720953592989622}{154786433280020486873} a^{15} - \frac{74609324324817045258}{154786433280020486873} a^{14} - \frac{15469327581241633802}{154786433280020486873} a^{13} + \frac{35600507997238845194}{154786433280020486873} a^{12} - \frac{58853820601336126532}{154786433280020486873} a^{11} - \frac{22588793237784827589}{154786433280020486873} a^{10} - \frac{581582023586344165}{3293328367660010359} a^{9} - \frac{64622701132867921900}{154786433280020486873} a^{8} + \frac{46491135760864079394}{154786433280020486873} a^{7} - \frac{45097091121473444800}{154786433280020486873} a^{6} - \frac{38500596827840246849}{154786433280020486873} a^{5} + \frac{66941311306586889409}{154786433280020486873} a^{4} + \frac{30738010108758734503}{154786433280020486873} a^{3} + \frac{1928540178966493510}{154786433280020486873} a^{2} + \frac{1519788557430540161}{154786433280020486873} a + \frac{65972366918879625353}{154786433280020486873}$

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:  $16$
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:  \( 109815062.576 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T28:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 22528
The 208 conjugacy class representatives for t22n28 are not computed
Character table for t22n28 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ $22$ R $22$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ $22$ 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
23Data not computed
47Data not computed