Properties

Label 22.0.298...003.1
Degree $22$
Signature $[0, 11]$
Discriminant $-2.989\times 10^{27}$
Root discriminant $17.74$
Ramified primes $3, 167$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{22}$ (as 22T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 4*x^20 + 3*x^19 + 11*x^18 - 6*x^17 - 18*x^16 + 2*x^15 + 26*x^14 - 8*x^13 - 18*x^12 + 23*x^11 + 12*x^10 - 27*x^9 - 30*x^8 + 23*x^7 + 19*x^6 + 3*x^5 + 5*x^4 - 26*x^3 + 8*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^22 - x^21 - 4*x^20 + 3*x^19 + 11*x^18 - 6*x^17 - 18*x^16 + 2*x^15 + 26*x^14 - 8*x^13 - 18*x^12 + 23*x^11 + 12*x^10 - 27*x^9 - 30*x^8 + 23*x^7 + 19*x^6 + 3*x^5 + 5*x^4 - 26*x^3 + 8*x^2 + 2*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 8, -26, 5, 3, 19, 23, -30, -27, 12, 23, -18, -8, 26, 2, -18, -6, 11, 3, -4, -1, 1]);
 

\( x^{22} - x^{21} - 4 x^{20} + 3 x^{19} + 11 x^{18} - 6 x^{17} - 18 x^{16} + 2 x^{15} + 26 x^{14} - 8 x^{13} - 18 x^{12} + 23 x^{11} + 12 x^{10} - 27 x^{9} - 30 x^{8} + 23 x^{7} + 19 x^{6} + 3 x^{5} + 5 x^{4} - 26 x^{3} + 8 x^{2} + 2 x + 1 \)

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:  \(-2988811416117414420033765003\)\(\medspace = -\,3^{11}\cdot 167^{10}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $17.74$
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:  $3, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{15} + \frac{1}{5} a^{14} + \frac{2}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{17} + \frac{2}{5} a^{15} - \frac{1}{5} a^{14} + \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{85} a^{18} + \frac{8}{85} a^{17} + \frac{6}{85} a^{16} + \frac{1}{5} a^{15} - \frac{1}{5} a^{14} + \frac{13}{85} a^{13} + \frac{12}{85} a^{12} - \frac{33}{85} a^{11} - \frac{31}{85} a^{9} + \frac{2}{85} a^{8} + \frac{19}{85} a^{7} - \frac{18}{85} a^{6} - \frac{31}{85} a^{5} + \frac{41}{85} a^{4} - \frac{41}{85} a^{3} + \frac{29}{85} a^{2} + \frac{6}{17} a + \frac{4}{17}$, $\frac{1}{85} a^{19} - \frac{7}{85} a^{17} + \frac{3}{85} a^{16} - \frac{2}{5} a^{15} - \frac{38}{85} a^{14} - \frac{7}{85} a^{13} + \frac{7}{85} a^{12} + \frac{9}{85} a^{11} + \frac{3}{85} a^{10} + \frac{12}{85} a^{9} + \frac{37}{85} a^{8} - \frac{2}{5} a^{7} - \frac{6}{85} a^{6} + \frac{22}{85} a^{4} + \frac{1}{5} a^{3} - \frac{32}{85} a^{2} + \frac{18}{85} a - \frac{41}{85}$, $\frac{1}{85} a^{20} + \frac{8}{85} a^{17} + \frac{8}{85} a^{16} - \frac{21}{85} a^{15} + \frac{2}{17} a^{14} - \frac{4}{85} a^{13} - \frac{26}{85} a^{12} - \frac{41}{85} a^{11} - \frac{39}{85} a^{10} + \frac{41}{85} a^{9} + \frac{31}{85} a^{8} + \frac{42}{85} a^{7} - \frac{24}{85} a^{6} - \frac{5}{17} a^{5} - \frac{19}{85} a^{4} - \frac{13}{85} a^{3} - \frac{18}{85} a - \frac{13}{85}$, $\frac{1}{10500305} a^{21} - \frac{1946}{617665} a^{20} + \frac{6791}{10500305} a^{19} + \frac{53459}{10500305} a^{18} + \frac{4252}{2100061} a^{17} + \frac{218369}{10500305} a^{16} - \frac{10972}{51221} a^{15} + \frac{3597096}{10500305} a^{14} + \frac{4053216}{10500305} a^{13} + \frac{15072}{617665} a^{12} - \frac{152688}{456535} a^{11} - \frac{128189}{456535} a^{10} + \frac{262911}{617665} a^{9} - \frac{1293033}{10500305} a^{8} + \frac{403505}{2100061} a^{7} + \frac{911804}{10500305} a^{6} - \frac{829627}{10500305} a^{5} + \frac{4011368}{10500305} a^{4} - \frac{1286802}{10500305} a^{3} + \frac{237819}{617665} a^{2} + \frac{206699}{617665} a + \frac{1171814}{10500305}$

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:  \( -\frac{252771}{2100061} a^{21} + \frac{1971991}{10500305} a^{20} + \frac{3815992}{10500305} a^{19} - \frac{7438603}{10500305} a^{18} - \frac{10147709}{10500305} a^{17} + \frac{3741150}{2100061} a^{16} + \frac{22072}{15065} a^{15} - \frac{23383672}{10500305} a^{14} - \frac{29887762}{10500305} a^{13} + \frac{40929768}{10500305} a^{12} + \frac{870837}{456535} a^{11} - \frac{2069523}{456535} a^{10} + \frac{1134992}{2100061} a^{9} + \frac{45113928}{10500305} a^{8} - \frac{3164372}{10500305} a^{7} - \frac{69379161}{10500305} a^{6} + \frac{263964}{10500305} a^{5} + \frac{32881823}{10500305} a^{4} + \frac{351687}{617665} a^{3} + \frac{36666409}{10500305} a^{2} - \frac{31325554}{10500305} a + \frac{2406049}{10500305} \) (order $6$)
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:  \( 138999.076948 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{11}\cdot 138999.076948 \cdot 1}{6\sqrt{2988811416117414420033765003}}\approx 0.255323013187$ (assuming GRH)

Galois group

$D_{22}$ (as 22T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 44
The 14 conjugacy class representatives for $D_{22}$
Character table for $D_{22}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.1.129891985607.1

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

Sibling fields

Galois closure: Deg 44
Degree 22 sibling: 22.2.499131506491608208145638755501.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{11}$ $22$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{11}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}$

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
3Data not computed
$167$167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$