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Properties

Label 22.6.53808401250...0000.1
Degree $22$
Signature $[6, 8]$
Discriminant $2^{36}\cdot 3^{16}\cdot 5^{39}$
Root discriminant $119.86$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T27

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21744, -91944, 82836, 564960, -1573820, 1599944, -682924, 7356, 146020, -196640, 301914, -302224, 156506, -18220, -26035, 14264, -504, -1424, 115, 120, -16, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 - 16*x^20 + 120*x^19 + 115*x^18 - 1424*x^17 - 504*x^16 + 14264*x^15 - 26035*x^14 - 18220*x^13 + 156506*x^12 - 302224*x^11 + 301914*x^10 - 196640*x^9 + 146020*x^8 + 7356*x^7 - 682924*x^6 + 1599944*x^5 - 1573820*x^4 + 564960*x^3 + 82836*x^2 - 91944*x + 21744)
 
gp: K = bnfinit(x^22 - 4*x^21 - 16*x^20 + 120*x^19 + 115*x^18 - 1424*x^17 - 504*x^16 + 14264*x^15 - 26035*x^14 - 18220*x^13 + 156506*x^12 - 302224*x^11 + 301914*x^10 - 196640*x^9 + 146020*x^8 + 7356*x^7 - 682924*x^6 + 1599944*x^5 - 1573820*x^4 + 564960*x^3 + 82836*x^2 - 91944*x + 21744, 1)
 

Normalized defining polynomial

\( x^{22} - 4 x^{21} - 16 x^{20} + 120 x^{19} + 115 x^{18} - 1424 x^{17} - 504 x^{16} + 14264 x^{15} - 26035 x^{14} - 18220 x^{13} + 156506 x^{12} - 302224 x^{11} + 301914 x^{10} - 196640 x^{9} + 146020 x^{8} + 7356 x^{7} - 682924 x^{6} + 1599944 x^{5} - 1573820 x^{4} + 564960 x^{3} + 82836 x^{2} - 91944 x + 21744 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5380840125000000000000000000000000000000000000=2^{36}\cdot 3^{16}\cdot 5^{39}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $119.86$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{15} a^{11} - \frac{2}{15} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{2}{15} a^{6} - \frac{1}{15} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{75} a^{12} + \frac{1}{75} a^{11} - \frac{16}{75} a^{10} + \frac{2}{15} a^{9} + \frac{4}{15} a^{8} + \frac{13}{75} a^{7} - \frac{37}{75} a^{6} + \frac{7}{75} a^{5} - \frac{2}{15} a^{4} + \frac{1}{15} a^{3} + \frac{2}{25} a^{2} + \frac{2}{25} a - \frac{12}{25}$, $\frac{1}{75} a^{13} - \frac{2}{75} a^{11} - \frac{4}{75} a^{10} + \frac{2}{15} a^{9} - \frac{7}{75} a^{8} + \frac{1}{3} a^{7} + \frac{14}{75} a^{6} - \frac{32}{75} a^{5} + \frac{1}{5} a^{4} + \frac{1}{75} a^{3} - \frac{9}{25} a + \frac{2}{25}$, $\frac{1}{150} a^{14} - \frac{1}{75} a^{11} + \frac{53}{150} a^{10} - \frac{31}{75} a^{9} - \frac{1}{15} a^{8} + \frac{4}{15} a^{7} - \frac{31}{150} a^{6} - \frac{23}{75} a^{5} + \frac{28}{75} a^{4} + \frac{1}{15} a^{3} + \frac{2}{5} a^{2} + \frac{3}{25} a - \frac{12}{25}$, $\frac{1}{150} a^{15} - \frac{1}{30} a^{11} + \frac{13}{75} a^{10} + \frac{1}{15} a^{9} - \frac{7}{15} a^{8} - \frac{1}{30} a^{7} - \frac{2}{15} a^{5} - \frac{1}{15} a^{4} + \frac{7}{15} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{8}{25}$, $\frac{1}{150} a^{16} - \frac{1}{150} a^{12} + \frac{1}{25} a^{10} - \frac{1}{5} a^{9} - \frac{1}{2} a^{8} + \frac{26}{75} a^{7} + \frac{7}{25} a^{6} + \frac{8}{25} a^{5} + \frac{1}{5} a^{4} + \frac{1}{3} a^{3} + \frac{9}{25} a^{2} + \frac{7}{25} a + \frac{11}{25}$, $\frac{1}{150} a^{17} - \frac{1}{150} a^{13} - \frac{2}{75} a^{11} - \frac{1}{15} a^{10} + \frac{1}{6} a^{9} - \frac{8}{25} a^{8} - \frac{4}{75} a^{7} + \frac{34}{75} a^{6} + \frac{4}{15} a^{5} - \frac{1}{3} a^{4} + \frac{2}{75} a^{3} - \frac{4}{75} a^{2} + \frac{1}{25} a - \frac{1}{5}$, $\frac{1}{300} a^{18} - \frac{1}{300} a^{14} - \frac{1}{150} a^{12} - \frac{2}{75} a^{11} - \frac{7}{300} a^{10} - \frac{7}{75} a^{9} + \frac{8}{75} a^{8} - \frac{14}{75} a^{7} - \frac{17}{150} a^{6} - \frac{3}{25} a^{5} + \frac{67}{150} a^{4} - \frac{37}{75} a^{3} - \frac{11}{25} a^{2} + \frac{11}{25} a - \frac{6}{25}$, $\frac{1}{4500} a^{19} + \frac{1}{1125} a^{18} + \frac{2}{1125} a^{17} - \frac{1}{375} a^{16} - \frac{11}{4500} a^{15} + \frac{1}{2250} a^{14} - \frac{11}{2250} a^{13} - \frac{2}{375} a^{12} - \frac{1}{500} a^{11} - \frac{511}{2250} a^{10} + \frac{472}{1125} a^{9} + \frac{203}{1125} a^{8} + \frac{82}{375} a^{7} - \frac{43}{2250} a^{6} - \frac{59}{2250} a^{5} - \frac{529}{1125} a^{4} - \frac{112}{375} a^{3} - \frac{14}{375} a^{2} + \frac{37}{125} a - \frac{9}{125}$, $\frac{1}{1381500} a^{20} + \frac{1}{92100} a^{19} - \frac{272}{345375} a^{18} - \frac{326}{345375} a^{17} - \frac{1583}{1381500} a^{16} + \frac{1021}{1381500} a^{15} - \frac{19}{9210} a^{14} + \frac{4307}{690750} a^{13} - \frac{1651}{460500} a^{12} + \frac{27799}{1381500} a^{11} + \frac{17588}{115125} a^{10} - \frac{32701}{69075} a^{9} - \frac{18431}{345375} a^{8} + \frac{5369}{690750} a^{7} + \frac{203033}{690750} a^{6} + \frac{11547}{76750} a^{5} - \frac{27571}{69075} a^{4} + \frac{15929}{115125} a^{3} - \frac{8496}{38375} a^{2} - \frac{9862}{38375} a - \frac{14084}{38375}$, $\frac{1}{1494410049269831572409888496962919000} a^{21} - \frac{17602644872363490516082009969}{49813668308994385746996283232097300} a^{20} + \frac{710256076182184567314156888159}{41511390257495321455830236026747750} a^{19} - \frac{17839814991083991229332028464427}{83022780514990642911660472053495500} a^{18} - \frac{521950548458397263654991435363097}{1494410049269831572409888496962919000} a^{17} - \frac{1920896030273897287711831843360639}{747205024634915786204944248481459500} a^{16} - \frac{1173005161082901327318644940670243}{373602512317457893102472124240729750} a^{15} + \frac{2147110597914907500788452717485199}{747205024634915786204944248481459500} a^{14} - \frac{6198777011365414425207329184944687}{1494410049269831572409888496962919000} a^{13} + \frac{2529106593943920489785361031792643}{747205024634915786204944248481459500} a^{12} - \frac{5548548059535412110633970083653327}{249068341544971928734981416160486500} a^{11} + \frac{90827072405614957349040491780768461}{249068341544971928734981416160486500} a^{10} - \frac{33918708663708219817146163115284241}{83022780514990642911660472053495500} a^{9} - \frac{106253536401938609444162926198780747}{373602512317457893102472124240729750} a^{8} - \frac{14994129448969423890010298420339549}{373602512317457893102472124240729750} a^{7} - \frac{71789606933961185739188020589371097}{186801256158728946551236062120364875} a^{6} - \frac{154311305680819838146202064590157769}{373602512317457893102472124240729750} a^{5} + \frac{130314625188961805078945085844597729}{373602512317457893102472124240729750} a^{4} - \frac{54945661939869758720541706472799797}{124534170772485964367490708080243250} a^{3} + \frac{30451036045459674815309243929736068}{62267085386242982183745354040121625} a^{2} - \frac{121288753319514034728255841011007}{8302278051499064291166047205349550} a - \frac{5816414875501130139678923342212359}{20755695128747660727915118013373875}$

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:  $13$
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:  \( 961054672237000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T27:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15840
The 20 conjugacy class representatives for t22n27
Character table for t22n27

Intermediate fields

\(\Q(\sqrt{5}) \), 11.3.6561000000000000000000.1

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

Sibling fields

Degree 22 sibling: data not computed
Degree 24 sibling: data not computed
Degree 44 sibling: 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 R R R $22$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.16.9$x^{8} + 2 x^{4} + 8 x + 12$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.20.34$x^{12} + 14 x^{10} + 16 x^{8} - 8 x^{6} - 8 x^{4} + 16 x^{2} + 16$$6$$2$$20$$S_4$$[8/3, 8/3]_{3}^{2}$
3Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.10.19.3$x^{10} + 30$$10$$1$$19$$F_5$$[9/4]_{4}$
5.10.19.3$x^{10} + 30$$10$$1$$19$$F_5$$[9/4]_{4}$