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Properties

Label 22.16.1781764468...3423.2
Degree $22$
Signature $[16, 3]$
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

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -9, 24, 289, -255, -2769, 2464, 8528, -7547, -10618, 10042, 5618, -6092, -1236, 1582, 266, -117, -163, 1, 52, -9, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 - 9*x^20 + 52*x^19 + x^18 - 163*x^17 - 117*x^16 + 266*x^15 + 1582*x^14 - 1236*x^13 - 6092*x^12 + 5618*x^11 + 10042*x^10 - 10618*x^9 - 7547*x^8 + 8528*x^7 + 2464*x^6 - 2769*x^5 - 255*x^4 + 289*x^3 + 24*x^2 - 9*x - 1)
 
gp: K = bnfinit(x^22 - 4*x^21 - 9*x^20 + 52*x^19 + x^18 - 163*x^17 - 117*x^16 + 266*x^15 + 1582*x^14 - 1236*x^13 - 6092*x^12 + 5618*x^11 + 10042*x^10 - 10618*x^9 - 7547*x^8 + 8528*x^7 + 2464*x^6 - 2769*x^5 - 255*x^4 + 289*x^3 + 24*x^2 - 9*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - 4 x^{21} - 9 x^{20} + 52 x^{19} + x^{18} - 163 x^{17} - 117 x^{16} + 266 x^{15} + 1582 x^{14} - 1236 x^{13} - 6092 x^{12} + 5618 x^{11} + 10042 x^{10} - 10618 x^{9} - 7547 x^{8} + 8528 x^{7} + 2464 x^{6} - 2769 x^{5} - 255 x^{4} + 289 x^{3} + 24 x^{2} - 9 x - 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 3]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{6046180221500885561049677} a^{21} - \frac{2792215067881754922808859}{6046180221500885561049677} a^{20} + \frac{1085325038480473510667048}{6046180221500885561049677} a^{19} + \frac{2801131952215091628741628}{6046180221500885561049677} a^{18} - \frac{2151913330059750272858426}{6046180221500885561049677} a^{17} - \frac{2350486801274375766642820}{6046180221500885561049677} a^{16} + \frac{2347343158468742011699711}{6046180221500885561049677} a^{15} + \frac{869610512467411831045567}{6046180221500885561049677} a^{14} - \frac{1542757314423613743023894}{6046180221500885561049677} a^{13} + \frac{2631683543492787223805299}{6046180221500885561049677} a^{12} + \frac{1240237749250371816046481}{6046180221500885561049677} a^{11} + \frac{2023870403528708952905501}{6046180221500885561049677} a^{10} - \frac{12837217854517827902645}{44132702346721792416421} a^{9} + \frac{363190592183562796643326}{6046180221500885561049677} a^{8} - \frac{2420625552650562281939502}{6046180221500885561049677} a^{7} + \frac{294359837634437826962645}{6046180221500885561049677} a^{6} - \frac{564023575187749693214579}{6046180221500885561049677} a^{5} - \frac{1196051991862565208148531}{6046180221500885561049677} a^{4} - \frac{1500093471001247937260599}{6046180221500885561049677} a^{3} - \frac{1902264009731662738383633}{6046180221500885561049677} a^{2} - \frac{2223211652217679993951103}{6046180221500885561049677} a + \frac{2356944316508829838086219}{6046180221500885561049677}$

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:  $18$
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:  \( 245612379.311 \) (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
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$