Properties

Label 22.0.310...627.1
Degree $22$
Signature $[0, 11]$
Discriminant $-3.102\times 10^{27}$
Root discriminant $17.77$
Ramified primes $3, 211441, 625831$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 22T47

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 4*x^20 - 2*x^18 + 9*x^17 - 4*x^16 + 6*x^15 + 4*x^14 - 2*x^13 + 10*x^12 + 7*x^10 + 4*x^9 + 2*x^8 + 11*x^7 + x^6 + 3*x^5 + 4*x^4 + 2*x^3 + 3*x^2 - x + 1)
 
gp: K = bnfinit(x^22 - 2*x^21 + 4*x^20 - 2*x^18 + 9*x^17 - 4*x^16 + 6*x^15 + 4*x^14 - 2*x^13 + 10*x^12 + 7*x^10 + 4*x^9 + 2*x^8 + 11*x^7 + x^6 + 3*x^5 + 4*x^4 + 2*x^3 + 3*x^2 - x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 3, 2, 4, 3, 1, 11, 2, 4, 7, 0, 10, -2, 4, 6, -4, 9, -2, 0, 4, -2, 1]);
 

\(x^{22} - 2 x^{21} + 4 x^{20} - 2 x^{18} + 9 x^{17} - 4 x^{16} + 6 x^{15} + 4 x^{14} - 2 x^{13} + 10 x^{12} + 7 x^{10} + 4 x^{9} + 2 x^{8} + 11 x^{7} + x^{6} + 3 x^{5} + 4 x^{4} + 2 x^{3} + 3 x^{2} - x + 1\)  Toggle raw display

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 11]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-3101889720909924494411835627\)\(\medspace = -\,3^{11}\cdot 211441^{2}\cdot 625831^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $17.77$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 211441, 625831$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{226307} a^{21} + \frac{58004}{226307} a^{20} + \frac{73859}{226307} a^{19} + \frac{47337}{226307} a^{18} + \frac{47189}{226307} a^{17} + \frac{61978}{226307} a^{16} - \frac{17138}{226307} a^{15} + \frac{59829}{226307} a^{14} + \frac{23133}{226307} a^{13} + \frac{78593}{226307} a^{12} - \frac{88947}{226307} a^{11} - \frac{112696}{226307} a^{10} + \frac{59833}{226307} a^{9} + \frac{28850}{226307} a^{8} - \frac{67163}{226307} a^{7} + \frac{18038}{226307} a^{6} + \frac{94968}{226307} a^{5} - \frac{51183}{226307} a^{4} + \frac{439}{226307} a^{3} - \frac{108055}{226307} a^{2} - \frac{39655}{226307} a - \frac{43583}{226307}$  Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $10$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{174996}{226307} a^{21} - \frac{306194}{226307} a^{20} + \frac{636794}{226307} a^{19} + \frac{44224}{226307} a^{18} - \frac{56186}{226307} a^{17} + \frac{1044341}{226307} a^{16} - \frac{61084}{226307} a^{15} + \frac{873864}{226307} a^{14} + \frac{455466}{226307} a^{13} + \frac{331624}{226307} a^{12} + \frac{931476}{226307} a^{11} + \frac{626913}{226307} a^{10} + \frac{1121234}{226307} a^{9} + \frac{630658}{226307} a^{8} + \frac{676618}{226307} a^{7} + \frac{1405654}{226307} a^{6} + \frac{844504}{226307} a^{5} + \frac{410792}{226307} a^{4} + \frac{331478}{226307} a^{3} + \frac{567526}{226307} a^{2} + \frac{237775}{226307} a + \frac{147846}{226307} \) (order $6$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 73023.7380593 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{11}\cdot 73023.7380593 \cdot 1}{6\sqrt{3101889720909924494411835627}}\approx 0.131667377430$ (assuming GRH)

Galois group

22T47:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 79833600
The 112 conjugacy class representatives for t22n47 are not computed
Character table for t22n47 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.5.132326332471.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 22 sibling: data not computed
Degree 44 sibling: 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 $22$ R $22$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ $22$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ $18{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
211441Data not computed
625831Data not computed