Properties

Label 22.0.152...375.1
Degree $22$
Signature $[0, 11]$
Discriminant $-1.526\times 10^{28}$
Root discriminant $19.10$
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 - 4*x^21 + 18*x^20 - 48*x^19 + 125*x^18 - 249*x^17 + 478*x^16 - 766*x^15 + 1216*x^14 - 1618*x^13 + 2242*x^12 - 2454*x^11 + 2995*x^10 - 2616*x^9 + 2692*x^8 - 1937*x^7 + 1684*x^6 - 1102*x^5 + 757*x^4 - 253*x^3 + 163*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^22 - 4*x^21 + 18*x^20 - 48*x^19 + 125*x^18 - 249*x^17 + 478*x^16 - 766*x^15 + 1216*x^14 - 1618*x^13 + 2242*x^12 - 2454*x^11 + 2995*x^10 - 2616*x^9 + 2692*x^8 - 1937*x^7 + 1684*x^6 - 1102*x^5 + 757*x^4 - 253*x^3 + 163*x^2 + 2*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 163, -253, 757, -1102, 1684, -1937, 2692, -2616, 2995, -2454, 2242, -1618, 1216, -766, 478, -249, 125, -48, 18, -4, 1]);
 

\( x^{22} - 4 x^{21} + 18 x^{20} - 48 x^{19} + 125 x^{18} - 249 x^{17} + 478 x^{16} - 766 x^{15} + 1216 x^{14} - 1618 x^{13} + 2242 x^{12} - 2454 x^{11} + 2995 x^{10} - 2616 x^{9} + 2692 x^{8} - 1937 x^{7} + 1684 x^{6} - 1102 x^{5} + 757 x^{4} - 253 x^{3} + 163 x^{2} + 2 x + 1 \)

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:  \(-15257581934366008831157764375\)\(\medspace = -\,5^{4}\cdot 7^{11}\cdot 83^{4}\cdot 127^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $19.10$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{73} a^{20} + \frac{6}{73} a^{19} + \frac{9}{73} a^{18} - \frac{7}{73} a^{17} + \frac{18}{73} a^{16} - \frac{24}{73} a^{15} + \frac{18}{73} a^{14} - \frac{25}{73} a^{13} + \frac{16}{73} a^{12} - \frac{25}{73} a^{11} + \frac{12}{73} a^{10} - \frac{25}{73} a^{9} + \frac{19}{73} a^{8} + \frac{29}{73} a^{7} - \frac{8}{73} a^{6} - \frac{3}{73} a^{5} + \frac{16}{73} a^{4} - \frac{5}{73} a^{3} - \frac{32}{73} a^{2} - \frac{9}{73} a + \frac{18}{73}$, $\frac{1}{895918078905504527250044171} a^{21} - \frac{1029675052787076496355520}{895918078905504527250044171} a^{20} + \frac{94541541919989199517743271}{895918078905504527250044171} a^{19} + \frac{3279287482768198647187820}{12272850395965815441781427} a^{18} + \frac{362414672188769874323791425}{895918078905504527250044171} a^{17} + \frac{16183252466243642279038080}{895918078905504527250044171} a^{16} + \frac{17652801120192881362439763}{895918078905504527250044171} a^{15} - \frac{304275821545373289596658293}{895918078905504527250044171} a^{14} - \frac{180145817138148519638314129}{895918078905504527250044171} a^{13} + \frac{421288684824846867665786979}{895918078905504527250044171} a^{12} - \frac{247757371849610266358468384}{895918078905504527250044171} a^{11} - \frac{34581351352840682837826610}{895918078905504527250044171} a^{10} + \frac{101403666819142800129172877}{895918078905504527250044171} a^{9} - \frac{70087196227760311472984377}{895918078905504527250044171} a^{8} + \frac{104871663917880612291542157}{895918078905504527250044171} a^{7} - \frac{7743340996765517892039431}{24214002132581203439190383} a^{6} - \frac{77467235939688095304313341}{895918078905504527250044171} a^{5} + \frac{149983118175567548776935831}{895918078905504527250044171} a^{4} - \frac{442206669760071561463669896}{895918078905504527250044171} a^{3} + \frac{119251387444657133110468054}{895918078905504527250044171} a^{2} + \frac{201870078523537954129136765}{895918078905504527250044171} a - \frac{185906182445383769053440881}{895918078905504527250044171}$

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:  \( -1 \) (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:  \( 69206.6233764 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{11}\cdot 69206.6233764 \cdot 1}{2\sqrt{15257581934366008831157764375}}\approx 0.168792583789$ (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

\(\Q(\sqrt{-7}) \), 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 ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ $22$ R R ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.4.2.2$x^{4} - 5 x^{2} + 50$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
7Data not computed
$83$83.2.0.1$x^{2} - x + 2$$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}$
127Data not computed