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Properties

Label 22.14.1855164453...2281.1
Degree $22$
Signature $[14, 4]$
Discriminant $23^{21}\cdot 47$
Root discriminant $23.76$
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, -8, 18, -6, -44, 283, -608, -196, 1683, -1021, -1009, 1655, -429, -1145, 627, 596, -283, -220, 81, 42, -14, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 14*x^20 + 42*x^19 + 81*x^18 - 220*x^17 - 283*x^16 + 596*x^15 + 627*x^14 - 1145*x^13 - 429*x^12 + 1655*x^11 - 1009*x^10 - 1021*x^9 + 1683*x^8 - 196*x^7 - 608*x^6 + 283*x^5 - 44*x^4 - 6*x^3 + 18*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^22 - 3*x^21 - 14*x^20 + 42*x^19 + 81*x^18 - 220*x^17 - 283*x^16 + 596*x^15 + 627*x^14 - 1145*x^13 - 429*x^12 + 1655*x^11 - 1009*x^10 - 1021*x^9 + 1683*x^8 - 196*x^7 - 608*x^6 + 283*x^5 - 44*x^4 - 6*x^3 + 18*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 3 x^{21} - 14 x^{20} + 42 x^{19} + 81 x^{18} - 220 x^{17} - 283 x^{16} + 596 x^{15} + 627 x^{14} - 1145 x^{13} - 429 x^{12} + 1655 x^{11} - 1009 x^{10} - 1021 x^{9} + 1683 x^{8} - 196 x^{7} - 608 x^{6} + 283 x^{5} - 44 x^{4} - 6 x^{3} + 18 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1855164453672687836700730712281=23^{21}\cdot 47\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.76$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{791842008232655903637809} a^{21} - \frac{135476618550820314916470}{791842008232655903637809} a^{20} + \frac{240182824973250868318837}{791842008232655903637809} a^{19} - \frac{72902056067384072356130}{791842008232655903637809} a^{18} + \frac{214700513235223518083273}{791842008232655903637809} a^{17} - \frac{253384012603032146992802}{791842008232655903637809} a^{16} - \frac{221833907176077125900692}{791842008232655903637809} a^{15} - \frac{364690714195646314846179}{791842008232655903637809} a^{14} + \frac{374346626598689858403034}{791842008232655903637809} a^{13} - \frac{160523787561681023552932}{791842008232655903637809} a^{12} + \frac{198313275494653137608}{16847702302822466034847} a^{11} + \frac{130681996231348939875111}{791842008232655903637809} a^{10} + \frac{272070730228804957727123}{791842008232655903637809} a^{9} + \frac{114308179982365840516987}{791842008232655903637809} a^{8} + \frac{6357667062988119851408}{16847702302822466034847} a^{7} + \frac{112223226818395730163764}{791842008232655903637809} a^{6} + \frac{209247643494875336394150}{791842008232655903637809} a^{5} - \frac{291072183375206533303352}{791842008232655903637809} a^{4} - \frac{389778713928263573828588}{791842008232655903637809} a^{3} - \frac{199423401525921717219298}{791842008232655903637809} a^{2} + \frac{220488909682540657178603}{791842008232655903637809} a - \frac{118089369565794375023600}{791842008232655903637809}$

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:  $17$
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:  \( 16297198.7643 \) (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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R $22$ $22$ $22$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ R $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.1$x^{2} - 47$$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$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$