Properties

Label 22.6.144...125.1
Degree $22$
Signature $[6, 8]$
Discriminant $1.447\times 10^{27}$
Root discriminant $17.16$
Ramified primes $5, 7, 83, 127$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group 22T46

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 - x^20 + 18*x^19 - 17*x^18 - 44*x^17 + 110*x^16 - 3*x^15 - 254*x^14 + 59*x^13 + 240*x^12 - 102*x^11 - 552*x^10 - 230*x^9 + 307*x^8 + 320*x^7 - 3*x^6 - 154*x^5 - 62*x^4 + 18*x^3 + 19*x^2 + 2*x - 1)
 
gp: K = bnfinit(x^22 - 2*x^21 - x^20 + 18*x^19 - 17*x^18 - 44*x^17 + 110*x^16 - 3*x^15 - 254*x^14 + 59*x^13 + 240*x^12 - 102*x^11 - 552*x^10 - 230*x^9 + 307*x^8 + 320*x^7 - 3*x^6 - 154*x^5 - 62*x^4 + 18*x^3 + 19*x^2 + 2*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 2, 19, 18, -62, -154, -3, 320, 307, -230, -552, -102, 240, 59, -254, -3, 110, -44, -17, 18, -1, -2, 1]);
 

\( x^{22} - 2 x^{21} - x^{20} + 18 x^{19} - 17 x^{18} - 44 x^{17} + 110 x^{16} - 3 x^{15} - 254 x^{14} + 59 x^{13} + 240 x^{12} - 102 x^{11} - 552 x^{10} - 230 x^{9} + 307 x^{8} + 320 x^{7} - 3 x^{6} - 154 x^{5} - 62 x^{4} + 18 x^{3} + 19 x^{2} + 2 x - 1 \)

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

Invariants

Degree:  $22$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[6, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(1447402975463751668017578125\)\(\medspace = 5^{11}\cdot 7^{4}\cdot 83^{4}\cdot 127^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $17.16$
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:  $5, 7, 83, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{2}{5} a^{17} - \frac{1}{5} a^{14} + \frac{2}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{19} + \frac{1}{5} a^{17} - \frac{1}{5} a^{15} + \frac{2}{5} a^{14} + \frac{2}{5} a^{13} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{20} - \frac{2}{5} a^{17} - \frac{1}{5} a^{16} + \frac{2}{5} a^{15} - \frac{2}{5} a^{14} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{56519985376604065507145} a^{21} - \frac{1332303350785640714766}{56519985376604065507145} a^{20} - \frac{3535365261660196536029}{56519985376604065507145} a^{19} - \frac{4919424422287081949599}{56519985376604065507145} a^{18} - \frac{18552319389979348044422}{56519985376604065507145} a^{17} - \frac{5480146535618417328722}{56519985376604065507145} a^{16} + \frac{2979282937667072444890}{11303997075320813101429} a^{15} - \frac{3979740765573833739454}{56519985376604065507145} a^{14} + \frac{2618457838077731340976}{56519985376604065507145} a^{13} + \frac{12954503807196842959579}{56519985376604065507145} a^{12} + \frac{18695186257425511168174}{56519985376604065507145} a^{11} + \frac{24886967222611872102271}{56519985376604065507145} a^{10} + \frac{23198866140311044023326}{56519985376604065507145} a^{9} - \frac{21710420089587276733561}{56519985376604065507145} a^{8} + \frac{4307330292767668606748}{11303997075320813101429} a^{7} - \frac{2365804169949362957548}{56519985376604065507145} a^{6} - \frac{20038793996407534264328}{56519985376604065507145} a^{5} - \frac{13162204338885356986351}{56519985376604065507145} a^{4} - \frac{8629938208056997339052}{56519985376604065507145} a^{3} + \frac{27563790421963051834119}{56519985376604065507145} a^{2} + \frac{582428447918856776305}{11303997075320813101429} a + \frac{10260943314429562231528}{56519985376604065507145}$

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:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 95631.3345105 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{6}\cdot(2\pi)^{8}\cdot 95631.3345105 \cdot 1}{2\sqrt{1447402975463751668017578125}}\approx 0.195386435924$ (assuming GRH)

Galois group

22T46:

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 39916800
The 62 conjugacy class representatives for t22n46 are not computed
Character table for t22n46 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 11.3.136113034225.1

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $22$ $22$ R R ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed
$83$83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.6.4.1$x^{6} + 415 x^{3} + 55112$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
83.10.0.1$x^{10} - x + 13$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$127$127.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
127.6.4.1$x^{6} + 1016 x^{3} + 435483$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
127.14.0.1$x^{14} - x + 29$$1$$14$$0$$C_{14}$$[\ ]^{14}$