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Properties

Label 22.12.1781764468...3423.5
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

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 9, -144, 174, 990, -368, -2547, -1329, 2624, 2895, -2463, -2216, 2227, 816, -1272, -58, 468, -184, -10, 16, 10, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 7*x^21 + 10*x^20 + 16*x^19 - 10*x^18 - 184*x^17 + 468*x^16 - 58*x^15 - 1272*x^14 + 816*x^13 + 2227*x^12 - 2216*x^11 - 2463*x^10 + 2895*x^9 + 2624*x^8 - 1329*x^7 - 2547*x^6 - 368*x^5 + 990*x^4 + 174*x^3 - 144*x^2 + 9*x + 1)
 
gp: K = bnfinit(x^22 - 7*x^21 + 10*x^20 + 16*x^19 - 10*x^18 - 184*x^17 + 468*x^16 - 58*x^15 - 1272*x^14 + 816*x^13 + 2227*x^12 - 2216*x^11 - 2463*x^10 + 2895*x^9 + 2624*x^8 - 1329*x^7 - 2547*x^6 - 368*x^5 + 990*x^4 + 174*x^3 - 144*x^2 + 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 7 x^{21} + 10 x^{20} + 16 x^{19} - 10 x^{18} - 184 x^{17} + 468 x^{16} - 58 x^{15} - 1272 x^{14} + 816 x^{13} + 2227 x^{12} - 2216 x^{11} - 2463 x^{10} + 2895 x^{9} + 2624 x^{8} - 1329 x^{7} - 2547 x^{6} - 368 x^{5} + 990 x^{4} + 174 x^{3} - 144 x^{2} + 9 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 5]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{4424101404174618201523345085363539} a^{21} - \frac{691382121286328625533238664103695}{4424101404174618201523345085363539} a^{20} + \frac{1041234757697913036561106350302376}{4424101404174618201523345085363539} a^{19} - \frac{1266452013592980192074412340045539}{4424101404174618201523345085363539} a^{18} - \frac{2105637670922554449697297216452030}{4424101404174618201523345085363539} a^{17} + \frac{188284558542529736726450996835421}{4424101404174618201523345085363539} a^{16} + \frac{986981131354783612138332428826684}{4424101404174618201523345085363539} a^{15} + \frac{256798703262186979700866914659574}{4424101404174618201523345085363539} a^{14} - \frac{183456965237564782698525792130215}{4424101404174618201523345085363539} a^{13} + \frac{39377276098097838915196917455249}{4424101404174618201523345085363539} a^{12} + \frac{1789055472657409696997834200270290}{4424101404174618201523345085363539} a^{11} - \frac{256445835042732163567316573389179}{4424101404174618201523345085363539} a^{10} - \frac{1086228336215554733512571073934921}{4424101404174618201523345085363539} a^{9} + \frac{748458587873644670262441657182658}{4424101404174618201523345085363539} a^{8} + \frac{754080560019927368543198534129699}{4424101404174618201523345085363539} a^{7} - \frac{1770759832750966974684942533841258}{4424101404174618201523345085363539} a^{6} - \frac{1047042955706215630626300697536496}{4424101404174618201523345085363539} a^{5} - \frac{1767151795772782233627004359641187}{4424101404174618201523345085363539} a^{4} - \frac{1497428498562335843477930539222430}{4424101404174618201523345085363539} a^{3} + \frac{729105122022644672518695785784905}{4424101404174618201523345085363539} a^{2} + \frac{974523279890585833199230873202296}{4424101404174618201523345085363539} a - \frac{1206824180367048953780211475236142}{4424101404174618201523345085363539}$

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:  \( 88229828.8508 \) (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