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Properties

Label 22.10.3790988231...9009.2
Degree $22$
Signature $[10, 6]$
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

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47, -59, -227, 160, 310, 309, -43, -1035, -467, 407, 1135, 353, -753, -520, 116, 368, 9, -150, 20, 30, -10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 - 10*x^20 + 30*x^19 + 20*x^18 - 150*x^17 + 9*x^16 + 368*x^15 + 116*x^14 - 520*x^13 - 753*x^12 + 353*x^11 + 1135*x^10 + 407*x^9 - 467*x^8 - 1035*x^7 - 43*x^6 + 309*x^5 + 310*x^4 + 160*x^3 - 227*x^2 - 59*x + 47)
 
gp: K = bnfinit(x^22 - 2*x^21 - 10*x^20 + 30*x^19 + 20*x^18 - 150*x^17 + 9*x^16 + 368*x^15 + 116*x^14 - 520*x^13 - 753*x^12 + 353*x^11 + 1135*x^10 + 407*x^9 - 467*x^8 - 1035*x^7 - 43*x^6 + 309*x^5 + 310*x^4 + 160*x^3 - 227*x^2 - 59*x + 47, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} - 10 x^{20} + 30 x^{19} + 20 x^{18} - 150 x^{17} + 9 x^{16} + 368 x^{15} + 116 x^{14} - 520 x^{13} - 753 x^{12} + 353 x^{11} + 1135 x^{10} + 407 x^{9} - 467 x^{8} - 1035 x^{7} - 43 x^{6} + 309 x^{5} + 310 x^{4} + 160 x^{3} - 227 x^{2} - 59 x + 47 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 6]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{14339434197834059665124564099} a^{21} - \frac{2840503473872223357949493024}{14339434197834059665124564099} a^{20} + \frac{835899927254228900733420341}{14339434197834059665124564099} a^{19} + \frac{6933864788607139226698472630}{14339434197834059665124564099} a^{18} + \frac{417306332388258496963505645}{14339434197834059665124564099} a^{17} + \frac{6148608126901336927915653605}{14339434197834059665124564099} a^{16} + \frac{6866005526815571107643723839}{14339434197834059665124564099} a^{15} - \frac{3777139134453603277554714209}{14339434197834059665124564099} a^{14} + \frac{5655146583668390787133256207}{14339434197834059665124564099} a^{13} - \frac{7120608924654383779123420583}{14339434197834059665124564099} a^{12} + \frac{5413570438986900414140091549}{14339434197834059665124564099} a^{11} + \frac{175743967500191279760803223}{14339434197834059665124564099} a^{10} + \frac{3419047979878358032314209129}{14339434197834059665124564099} a^{9} - \frac{3483820307953559755414071246}{14339434197834059665124564099} a^{8} - \frac{72033775511189654088748990}{14339434197834059665124564099} a^{7} + \frac{5657790765249753008278471303}{14339434197834059665124564099} a^{6} + \frac{4463845631202412159154456373}{14339434197834059665124564099} a^{5} - \frac{4091631248874404578636658041}{14339434197834059665124564099} a^{4} + \frac{4993249914970568472235887478}{14339434197834059665124564099} a^{3} - \frac{3337890984795171029385975362}{14339434197834059665124564099} a^{2} - \frac{1380290903619700514091498227}{14339434197834059665124564099} a - \frac{66699439205511033769451337}{305094344634767226917543917}$

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:  $15$
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:  \( 10678633.8472 \) (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}$
$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}$
47.2.1.1$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}$