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Properties

Label 22.22.3911379304...5633.1
Degree $22$
Signature $[22, 0]$
Discriminant $3^{10}\cdot 11^{32}\cdot 73^{11}$
Root discriminant $460.56$
Ramified primes $3, 11, 73$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T11

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-561087, 347490, 7144929, -579150, -23816430, 270270, 30961359, -51480, -20640906, 4290, 8027019, -130, -1945944, 0, 302940, 0, -30294, 0, 1881, 0, -66, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 66*x^20 + 1881*x^18 - 30294*x^16 + 302940*x^14 - 1945944*x^12 - 130*x^11 + 8027019*x^10 + 4290*x^9 - 20640906*x^8 - 51480*x^7 + 30961359*x^6 + 270270*x^5 - 23816430*x^4 - 579150*x^3 + 7144929*x^2 + 347490*x - 561087)
 
gp: K = bnfinit(x^22 - 66*x^20 + 1881*x^18 - 30294*x^16 + 302940*x^14 - 1945944*x^12 - 130*x^11 + 8027019*x^10 + 4290*x^9 - 20640906*x^8 - 51480*x^7 + 30961359*x^6 + 270270*x^5 - 23816430*x^4 - 579150*x^3 + 7144929*x^2 + 347490*x - 561087, 1)
 

Normalized defining polynomial

\( x^{22} - 66 x^{20} + 1881 x^{18} - 30294 x^{16} + 302940 x^{14} - 1945944 x^{12} - 130 x^{11} + 8027019 x^{10} + 4290 x^{9} - 20640906 x^{8} - 51480 x^{7} + 30961359 x^{6} + 270270 x^{5} - 23816430 x^{4} - 579150 x^{3} + 7144929 x^{2} + 347490 x - 561087 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(39113793049554550639626744281864455524059155087546195005633=3^{10}\cdot 11^{32}\cdot 73^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $460.56$
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:  $3, 11, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{176} a^{11} - \frac{3}{16} a^{9} + \frac{1}{4} a^{7} + \frac{3}{16} a^{5} + \frac{5}{16} a^{3} - \frac{3}{16} a + \frac{23}{176}$, $\frac{1}{176} a^{12} - \frac{3}{16} a^{10} + \frac{1}{4} a^{8} + \frac{3}{16} a^{6} + \frac{5}{16} a^{4} - \frac{3}{16} a^{2} + \frac{23}{176} a$, $\frac{1}{176} a^{13} + \frac{1}{16} a^{9} + \frac{7}{16} a^{7} - \frac{1}{2} a^{5} + \frac{1}{8} a^{3} + \frac{23}{176} a^{2} - \frac{3}{16} a + \frac{5}{16}$, $\frac{1}{528} a^{14} - \frac{5}{16} a^{10} - \frac{3}{16} a^{8} - \frac{1}{2} a^{6} + \frac{3}{8} a^{4} + \frac{23}{528} a^{3} - \frac{1}{16} a^{2} + \frac{7}{16} a$, $\frac{1}{528} a^{15} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} - \frac{5}{16} a^{5} + \frac{23}{528} a^{4} + \frac{1}{8} a^{3} + \frac{7}{16} a^{2} - \frac{5}{16} a + \frac{3}{16}$, $\frac{1}{1584} a^{16} - \frac{1}{2} a^{10} - \frac{1}{4} a^{8} - \frac{7}{16} a^{6} + \frac{23}{1584} a^{5} + \frac{3}{8} a^{4} + \frac{23}{48} a^{3} - \frac{7}{16} a^{2} + \frac{1}{16} a$, $\frac{1}{58608} a^{17} - \frac{5}{19536} a^{16} - \frac{17}{19536} a^{15} + \frac{3}{3256} a^{14} + \frac{1}{814} a^{13} - \frac{3}{3256} a^{12} + \frac{3}{3256} a^{11} + \frac{16}{37} a^{10} + \frac{105}{296} a^{9} - \frac{17}{296} a^{8} + \frac{21}{592} a^{7} + \frac{4931}{14652} a^{6} + \frac{1873}{9768} a^{5} + \frac{527}{3256} a^{4} + \frac{75}{814} a^{3} + \frac{217}{6512} a^{2} + \frac{499}{1628} a + \frac{2217}{6512}$, $\frac{1}{175824} a^{18} - \frac{1}{3256} a^{16} - \frac{5}{19536} a^{15} - \frac{13}{19536} a^{14} + \frac{1}{6512} a^{13} + \frac{9}{6512} a^{12} - \frac{9}{3256} a^{11} + \frac{9}{296} a^{10} + \frac{15}{592} a^{9} + \frac{11}{296} a^{8} - \frac{27659}{87912} a^{7} - \frac{261}{592} a^{6} - \frac{1526}{3663} a^{5} - \frac{9157}{19536} a^{4} - \frac{1183}{4884} a^{3} + \frac{1535}{3256} a^{2} + \frac{247}{1628} a - \frac{87}{1628}$, $\frac{1}{175824} a^{19} + \frac{1}{5328} a^{16} + \frac{7}{9768} a^{15} - \frac{1}{3256} a^{14} + \frac{5}{6512} a^{13} - \frac{15}{6512} a^{12} + \frac{5}{3256} a^{11} + \frac{35}{592} a^{10} + \frac{51}{296} a^{9} + \frac{15665}{175824} a^{8} - \frac{179}{592} a^{7} + \frac{1085}{5328} a^{6} - \frac{569}{2664} a^{5} - \frac{457}{1221} a^{4} - \frac{5935}{19536} a^{3} - \frac{2173}{6512} a^{2} - \frac{939}{6512} a - \frac{3125}{6512}$, $\frac{1}{527472} a^{20} - \frac{5}{58608} a^{16} + \frac{1}{1776} a^{15} + \frac{13}{19536} a^{14} + \frac{15}{6512} a^{13} + \frac{13}{6512} a^{12} - \frac{17}{6512} a^{11} - \frac{251}{592} a^{10} + \frac{54869}{527472} a^{9} + \frac{287}{592} a^{8} - \frac{2495}{7992} a^{7} + \frac{103}{592} a^{6} + \frac{873}{3256} a^{5} + \frac{127}{888} a^{4} + \frac{5953}{19536} a^{3} - \frac{2823}{6512} a^{2} - \frac{1473}{3256} a + \frac{973}{6512}$, $\frac{1}{527472} a^{21} - \frac{5}{58608} a^{16} + \frac{1}{9768} a^{15} - \frac{13}{19536} a^{14} + \frac{1}{407} a^{13} - \frac{5}{3256} a^{12} + \frac{7}{6512} a^{11} - \frac{22591}{131868} a^{10} + \frac{95}{296} a^{9} + \frac{301}{1998} a^{8} - \frac{51}{592} a^{7} - \frac{797}{2664} a^{6} - \frac{7837}{58608} a^{5} + \frac{949}{2442} a^{4} - \frac{2041}{4884} a^{3} + \frac{34}{407} a^{2} - \frac{305}{1628} a + \frac{1417}{3256}$

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:  $21$
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:  \( 189982300459000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T11:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1210
The 25 conjugacy class representatives for t22n11
Character table for t22n11 is not computed

Intermediate fields

\(\Q(\sqrt{73}) \)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.11.10.1$x^{11} - 3$$11$$1$$10$$C_{11}:C_5$$[\ ]_{11}^{5}$
11Data not computed
73Data not computed