↑

Properties

Label 22.10.3790988231...9009.1
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

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 14, 16, -180, 144, 381, -712, -5, 986, -541, -569, 571, 86, -426, 302, 41, -208, 128, -6, -42, 30, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 9*x^21 + 30*x^20 - 42*x^19 - 6*x^18 + 128*x^17 - 208*x^16 + 41*x^15 + 302*x^14 - 426*x^13 + 86*x^12 + 571*x^11 - 569*x^10 - 541*x^9 + 986*x^8 - 5*x^7 - 712*x^6 + 381*x^5 + 144*x^4 - 180*x^3 + 16*x^2 + 14*x - 1)
 
gp: K = bnfinit(x^22 - 9*x^21 + 30*x^20 - 42*x^19 - 6*x^18 + 128*x^17 - 208*x^16 + 41*x^15 + 302*x^14 - 426*x^13 + 86*x^12 + 571*x^11 - 569*x^10 - 541*x^9 + 986*x^8 - 5*x^7 - 712*x^6 + 381*x^5 + 144*x^4 - 180*x^3 + 16*x^2 + 14*x - 1, 1)
 

Normalized defining polynomial

\( x^{22} - 9 x^{21} + 30 x^{20} - 42 x^{19} - 6 x^{18} + 128 x^{17} - 208 x^{16} + 41 x^{15} + 302 x^{14} - 426 x^{13} + 86 x^{12} + 571 x^{11} - 569 x^{10} - 541 x^{9} + 986 x^{8} - 5 x^{7} - 712 x^{6} + 381 x^{5} + 144 x^{4} - 180 x^{3} + 16 x^{2} + 14 x - 1 \)

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

Invariants

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{796303717831977260992571} a^{21} - \frac{87869295388770387052784}{796303717831977260992571} a^{20} - \frac{283514081670568958242846}{796303717831977260992571} a^{19} + \frac{370258813279094025694540}{796303717831977260992571} a^{18} - \frac{376766871624346283264455}{796303717831977260992571} a^{17} - \frac{336693188591419385393601}{796303717831977260992571} a^{16} + \frac{277072826253642572000119}{796303717831977260992571} a^{15} + \frac{389910124695753150384049}{796303717831977260992571} a^{14} + \frac{294376130154104727872357}{796303717831977260992571} a^{13} + \frac{337742760289180585975073}{796303717831977260992571} a^{12} - \frac{179169175531159185299291}{796303717831977260992571} a^{11} - \frac{281825175719438843839906}{796303717831977260992571} a^{10} - \frac{47915328220971669142337}{796303717831977260992571} a^{9} + \frac{250606185278566657592757}{796303717831977260992571} a^{8} + \frac{98517648474582874042333}{796303717831977260992571} a^{7} - \frac{291126173066847932179237}{796303717831977260992571} a^{6} + \frac{152270661227022662724141}{796303717831977260992571} a^{5} - \frac{191758082522646565530917}{796303717831977260992571} a^{4} + \frac{135173530963357933749379}{796303717831977260992571} a^{3} - \frac{219902031222950776200947}{796303717831977260992571} a^{2} + \frac{268353336622270446209341}{796303717831977260992571} a + \frac{270952279619142597697668}{796303717831977260992571}$

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:  \( 10146146.0999 \) (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$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$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$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$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}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$