↑

Properties

Label 22.2.15029499322...0625.1
Degree $22$
Signature $[2, 10]$
Discriminant $5^{16}\cdot 11^{24}$
Root discriminant $44.10$
Ramified primes $5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T18

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12379, 12419, 121, -62095, -1210, 86933, 4719, -49676, -9438, 12419, 11011, -1129, -8008, 0, 3740, 0, -1122, 0, 209, 0, -22, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 1129*x^11 + 11011*x^10 + 12419*x^9 - 9438*x^8 - 49676*x^7 + 4719*x^6 + 86933*x^5 - 1210*x^4 - 62095*x^3 + 121*x^2 + 12419*x + 12379)
 
gp: K = bnfinit(x^22 - 22*x^20 + 209*x^18 - 1122*x^16 + 3740*x^14 - 8008*x^12 - 1129*x^11 + 11011*x^10 + 12419*x^9 - 9438*x^8 - 49676*x^7 + 4719*x^6 + 86933*x^5 - 1210*x^4 - 62095*x^3 + 121*x^2 + 12419*x + 12379, 1)
 

Normalized defining polynomial

\( x^{22} - 22 x^{20} + 209 x^{18} - 1122 x^{16} + 3740 x^{14} - 8008 x^{12} - 1129 x^{11} + 11011 x^{10} + 12419 x^{9} - 9438 x^{8} - 49676 x^{7} + 4719 x^{6} + 86933 x^{5} - 1210 x^{4} - 62095 x^{3} + 121 x^{2} + 12419 x + 12379 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1502949932221620345262426910400390625=5^{16}\cdot 11^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.10$
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:  $5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{495} a^{11} - \frac{1}{45} a^{9} + \frac{4}{45} a^{7} - \frac{7}{45} a^{5} + \frac{1}{9} a^{3} - \frac{1}{45} a + \frac{178}{495}$, $\frac{1}{495} a^{12} - \frac{1}{45} a^{10} + \frac{4}{45} a^{8} - \frac{7}{45} a^{6} + \frac{1}{9} a^{4} - \frac{1}{45} a^{2} + \frac{178}{495} a$, $\frac{1}{495} a^{13} - \frac{7}{45} a^{9} + \frac{7}{45} a^{7} + \frac{1}{15} a^{5} - \frac{1}{3} a^{4} - \frac{2}{15} a^{3} - \frac{152}{495} a^{2} + \frac{19}{45} a + \frac{13}{45}$, $\frac{1}{495} a^{14} - \frac{7}{45} a^{10} + \frac{7}{45} a^{8} + \frac{1}{15} a^{6} - \frac{1}{3} a^{5} - \frac{2}{15} a^{4} - \frac{152}{495} a^{3} + \frac{19}{45} a^{2} + \frac{13}{45} a$, $\frac{1}{495} a^{15} + \frac{1}{9} a^{9} - \frac{4}{45} a^{7} - \frac{4}{9} a^{5} - \frac{152}{495} a^{4} - \frac{16}{45} a^{3} - \frac{17}{45} a^{2} - \frac{17}{45} a + \frac{16}{45}$, $\frac{1}{5445} a^{16} - \frac{1}{5445} a^{15} - \frac{4}{5445} a^{14} + \frac{1}{1815} a^{13} + \frac{1}{1815} a^{12} - \frac{1}{5445} a^{11} - \frac{1}{33} a^{10} + \frac{4}{99} a^{9} - \frac{13}{99} a^{8} + \frac{4}{55} a^{7} - \frac{53}{495} a^{6} - \frac{1241}{5445} a^{5} + \frac{49}{363} a^{4} + \frac{2159}{5445} a^{3} - \frac{133}{1089} a^{2} + \frac{266}{1815} a - \frac{116}{363}$, $\frac{1}{81675} a^{17} - \frac{1}{81675} a^{16} + \frac{73}{81675} a^{15} - \frac{74}{81675} a^{14} + \frac{14}{81675} a^{13} + \frac{76}{81675} a^{12} - \frac{2}{7425} a^{11} + \frac{482}{7425} a^{10} - \frac{29}{297} a^{9} - \frac{13}{495} a^{8} - \frac{289}{2475} a^{7} - \frac{212}{27225} a^{6} + \frac{12829}{27225} a^{5} + \frac{10814}{27225} a^{4} - \frac{4508}{27225} a^{3} - \frac{3173}{81675} a^{2} - \frac{39943}{81675} a + \frac{2699}{7425}$, $\frac{1}{81675} a^{18} - \frac{1}{27225} a^{16} + \frac{74}{81675} a^{15} + \frac{1}{1089} a^{14} + \frac{2}{5445} a^{13} - \frac{2}{27225} a^{12} + \frac{1}{1089} a^{11} - \frac{67}{825} a^{10} - \frac{14}{135} a^{9} + \frac{281}{2475} a^{8} + \frac{1418}{9075} a^{7} + \frac{272}{2475} a^{6} + \frac{1167}{3025} a^{5} + \frac{12736}{27225} a^{4} + \frac{7993}{81675} a^{3} + \frac{1002}{3025} a^{2} + \frac{337}{27225} a + \frac{33469}{81675}$, $\frac{1}{898425} a^{19} + \frac{1}{179685} a^{18} + \frac{2}{898425} a^{17} - \frac{4}{99825} a^{16} + \frac{49}{59895} a^{15} - \frac{152}{179685} a^{14} + \frac{769}{898425} a^{13} - \frac{68}{179685} a^{12} + \frac{487}{898425} a^{11} + \frac{1216}{16335} a^{10} - \frac{9422}{81675} a^{9} + \frac{9754}{299475} a^{8} + \frac{10402}{299475} a^{7} - \frac{39782}{299475} a^{6} - \frac{34603}{99825} a^{5} - \frac{39197}{898425} a^{4} - \frac{298861}{898425} a^{3} + \frac{339821}{898425} a^{2} + \frac{50954}{898425} a + \frac{69959}{179685}$, $\frac{1}{898425} a^{20} - \frac{1}{898425} a^{18} - \frac{2}{898425} a^{17} - \frac{4}{179685} a^{16} - \frac{13}{19965} a^{15} + \frac{818}{898425} a^{14} + \frac{61}{898425} a^{13} - \frac{706}{898425} a^{12} + \frac{68}{99825} a^{11} - \frac{997}{16335} a^{10} + \frac{28382}{898425} a^{9} + \frac{164}{27225} a^{8} - \frac{1627}{11979} a^{7} + \frac{45722}{299475} a^{6} + \frac{356404}{898425} a^{5} - \frac{3776}{33275} a^{4} + \frac{421451}{898425} a^{3} - \frac{8929}{59895} a^{2} + \frac{10786}{35937} a - \frac{158441}{898425}$, $\frac{1}{898425} a^{21} + \frac{1}{299475} a^{18} + \frac{4}{898425} a^{17} + \frac{17}{898425} a^{16} + \frac{76}{99825} a^{15} + \frac{478}{898425} a^{14} + \frac{536}{898425} a^{13} + \frac{98}{299475} a^{12} + \frac{773}{898425} a^{11} + \frac{23146}{898425} a^{10} - \frac{1973}{16335} a^{9} - \frac{1601}{27225} a^{8} - \frac{19864}{299475} a^{7} - \frac{1994}{898425} a^{6} - \frac{827}{3993} a^{5} + \frac{133171}{299475} a^{4} - \frac{192919}{898425} a^{3} + \frac{19427}{179685} a^{2} - \frac{87478}{898425} a - \frac{292462}{898425}$

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:  $11$
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:  \( 3420605177.44 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T18:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2420
The 26 conjugacy class representatives for t22n18
Character table for t22n18 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \)

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 $20{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ R $20{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ R $20{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $20{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ $20{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
5Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.11.16.4$x^{11} + 22 x^{6} + 11$$11$$1$$16$$C_{11}:C_5$$[8/5]_{5}$