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Properties

Label 22.6.42787299364...7888.1
Degree $22$
Signature $[6, 8]$
Discriminant $2^{28}\cdot 3^{29}\cdot 11^{33}$
Root discriminant $375.10$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T40

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-73875456, -343056384, 2415638016, 2102474880, -3841506240, 378541152, 357855696, -756792960, 706449216, -597596736, 89570008, -101103984, 6526476, -5020334, 1841301, 39402, 82500, -9042, -462, -506, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 506*x^19 - 462*x^18 - 9042*x^17 + 82500*x^16 + 39402*x^15 + 1841301*x^14 - 5020334*x^13 + 6526476*x^12 - 101103984*x^11 + 89570008*x^10 - 597596736*x^9 + 706449216*x^8 - 756792960*x^7 + 357855696*x^6 + 378541152*x^5 - 3841506240*x^4 + 2102474880*x^3 + 2415638016*x^2 - 343056384*x - 73875456)
 
gp: K = bnfinit(x^22 - 506*x^19 - 462*x^18 - 9042*x^17 + 82500*x^16 + 39402*x^15 + 1841301*x^14 - 5020334*x^13 + 6526476*x^12 - 101103984*x^11 + 89570008*x^10 - 597596736*x^9 + 706449216*x^8 - 756792960*x^7 + 357855696*x^6 + 378541152*x^5 - 3841506240*x^4 + 2102474880*x^3 + 2415638016*x^2 - 343056384*x - 73875456, 1)
 

Normalized defining polynomial

\( x^{22} - 506 x^{19} - 462 x^{18} - 9042 x^{17} + 82500 x^{16} + 39402 x^{15} + 1841301 x^{14} - 5020334 x^{13} + 6526476 x^{12} - 101103984 x^{11} + 89570008 x^{10} - 597596736 x^{9} + 706449216 x^{8} - 756792960 x^{7} + 357855696 x^{6} + 378541152 x^{5} - 3841506240 x^{4} + 2102474880 x^{3} + 2415638016 x^{2} - 343056384 x - 73875456 \)

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:  \(427872993643659401569827455066929935858397318006217637888=2^{28}\cdot 3^{29}\cdot 11^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $375.10$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{6} - \frac{3}{16} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{32} a^{7} + \frac{1}{32} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{96} a^{12} - \frac{1}{32} a^{10} + \frac{1}{48} a^{9} + \frac{1}{32} a^{8} - \frac{1}{8} a^{7} - \frac{5}{96} a^{6} + \frac{3}{16} a^{5} - \frac{1}{4} a^{4} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{384} a^{13} - \frac{1}{192} a^{12} - \frac{1}{128} a^{11} - \frac{5}{192} a^{10} - \frac{1}{384} a^{9} + \frac{3}{64} a^{8} + \frac{31}{384} a^{7} + \frac{5}{192} a^{6} + \frac{3}{16} a^{5} + \frac{1}{16} a^{4} + \frac{9}{32} a^{3} - \frac{3}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{384} a^{14} + \frac{1}{384} a^{12} - \frac{1}{96} a^{11} - \frac{3}{128} a^{10} + \frac{1}{48} a^{9} - \frac{5}{384} a^{8} - \frac{3}{32} a^{7} - \frac{1}{12} a^{6} + \frac{5}{32} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{2304} a^{15} - \frac{1}{1152} a^{14} - \frac{1}{768} a^{13} + \frac{1}{1152} a^{12} - \frac{13}{2304} a^{11} + \frac{11}{384} a^{10} - \frac{65}{2304} a^{9} - \frac{25}{1152} a^{8} + \frac{23}{192} a^{7} - \frac{5}{96} a^{6} + \frac{29}{192} a^{5} - \frac{3}{32} a^{4} - \frac{23}{48} a^{3} + \frac{1}{12} a^{2} + \frac{1}{4} a$, $\frac{1}{2304} a^{16} - \frac{1}{2304} a^{14} + \frac{1}{1152} a^{13} + \frac{1}{256} a^{12} - \frac{1}{1152} a^{11} + \frac{25}{2304} a^{10} + \frac{3}{128} a^{9} + \frac{19}{1152} a^{8} - \frac{29}{384} a^{7} - \frac{1}{8} a^{6} - \frac{11}{48} a^{5} - \frac{7}{96} a^{4} - \frac{15}{32} a^{3} + \frac{17}{48} a^{2} + \frac{1}{4} a$, $\frac{1}{2304} a^{17} - \frac{1}{192} a^{12} + \frac{5}{384} a^{11} - \frac{1}{64} a^{10} - \frac{23}{768} a^{9} - \frac{29}{576} a^{8} + \frac{5}{128} a^{7} + \frac{23}{192} a^{6} + \frac{13}{64} a^{5} + \frac{3}{32} a^{3} + \frac{7}{48} a^{2} + \frac{1}{4} a$, $\frac{1}{9216} a^{18} - \frac{1}{4608} a^{16} - \frac{1}{4608} a^{15} - \frac{1}{1536} a^{14} + \frac{1}{4608} a^{13} - \frac{7}{2304} a^{12} - \frac{23}{1536} a^{11} + \frac{169}{9216} a^{10} - \frac{101}{4608} a^{9} + \frac{17}{1536} a^{8} + \frac{85}{768} a^{7} - \frac{79}{768} a^{6} - \frac{7}{128} a^{5} + \frac{35}{384} a^{4} + \frac{41}{192} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{18432} a^{19} + \frac{1}{9216} a^{17} - \frac{1}{9216} a^{16} - \frac{1}{9216} a^{15} + \frac{1}{1024} a^{14} - \frac{1}{1152} a^{13} + \frac{43}{9216} a^{12} - \frac{73}{6144} a^{11} + \frac{19}{9216} a^{10} - \frac{133}{9216} a^{9} - \frac{193}{4608} a^{8} + \frac{51}{512} a^{7} - \frac{89}{768} a^{6} + \frac{17}{256} a^{5} + \frac{65}{384} a^{4} + \frac{13}{96} a^{3} + \frac{19}{48} a^{2} + \frac{1}{4} a$, $\frac{1}{36864} a^{20} - \frac{1}{18432} a^{18} + \frac{1}{6144} a^{17} + \frac{1}{6144} a^{16} + \frac{1}{18432} a^{15} - \frac{5}{4608} a^{14} + \frac{1}{6144} a^{13} + \frac{133}{36864} a^{12} - \frac{173}{18432} a^{11} + \frac{11}{6144} a^{10} + \frac{67}{3072} a^{9} - \frac{535}{9216} a^{8} + \frac{125}{1536} a^{7} + \frac{113}{1536} a^{6} - \frac{181}{768} a^{5} + \frac{15}{128} a^{4} - \frac{19}{64} a^{3} + \frac{7}{24} a^{2} + \frac{1}{4} a$, $\frac{1}{681246073937297740407674507544975667487050935270779909008897133139825450221568} a^{21} + \frac{459268922172199235257380109278425234972509981226751556667843264991545929}{37847004107627652244870805974720870415947274181709994944938729618879191678976} a^{20} + \frac{284792410331882641733147057263387597215682385766721425648753487232579273}{18923502053813826122435402987360435207973637090854997472469364809439595839488} a^{19} - \frac{16254132648056447058160703311948325879687171552345319609740187202134470377}{340623036968648870203837253772487833743525467635389954504448566569912725110784} a^{18} - \frac{24482122929268931682629036088231792722878397051659376272993636252740864211}{113541012322882956734612417924162611247841822545129984834816188856637575036928} a^{17} + \frac{2579481459689178885950679412118599156693983204918328925141350010812468177}{16220144617554708104944631132023230178263117506447140690688026979519653576704} a^{16} - \frac{1535263562749283957051493720800709975212190463975735750487748074072013527}{28385253080720739183653104481040652811960455636282496208704047214159393759232} a^{15} - \frac{78061890066647856806586227645525775357216926889370934939983723393702315633}{113541012322882956734612417924162611247841822545129984834816188856637575036928} a^{14} - \frac{245606660202259100141175333729456453856671951966786034333998535549842661949}{227082024645765913469224835848325222495683645090259969669632377713275150073856} a^{13} + \frac{532259301347171418772271161267305812643528468064787335069904173408403611969}{170311518484324435101918626886243916871762733817694977252224283284956362555392} a^{12} + \frac{106669331423146505743740136676855542405198868037227700829664464582077545495}{56770506161441478367306208962081305623920911272564992417408094428318787518464} a^{11} + \frac{24332735269560976375465276357079744128738391195243400584406259410059404873}{28385253080720739183653104481040652811960455636282496208704047214159393759232} a^{10} - \frac{1822799582866848024937887963515130687797187850096985329734646128547203280639}{85155759242162217550959313443121958435881366908847488626112141642478181277696} a^{9} + \frac{119862388543029740655895421766779932672084669574880984823318730168421538061}{2027518077194338513118078891502903772282889688305892586336003372439956697088} a^{8} + \frac{122744083181086673439346002029384212376353528285614996735916652709626054995}{2365437756726728265304425373420054400996704636356874684058670601179949479936} a^{7} + \frac{407322777420049395548007154305870986400412259656465041730040171842969596409}{3548156635090092397956638060130081601495056954535312026088005901769924219904} a^{6} - \frac{712427775413088975917759093070611033120422204667689315819639885277840632403}{4730875513453456530608850746840108801993409272713749368117341202359898959872} a^{5} - \frac{2423152114673685846244008175135360749149948754546083482636420133186944979}{73919929897710258290763292919376700031147019886152333876833456286873421248} a^{4} + \frac{51779947223129749437611505873480267992636129748348821627255122984765694423}{122350228796210082688159933107933848327415757052941794003034686267928421376} a^{3} - \frac{27739202011102783459631998667858494426369356554617143582919880791387388745}{98559906530280344387684390559168933374862693181536445169111275049164561664} a^{2} - \frac{170487111924306569045327788549951025718655074841839590200849637923639975}{1319998748173397469477915944988869643413339640824148819229168862265596808} a - \frac{1341000645151089379725539082456494310799811677764875788053012855538959617}{3079997079071260762115137204974029167964459161923013911534727345286392552}$

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:  \( 162785860272000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T40:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 871200
The 44 conjugacy class representatives for t22n40
Character table for t22n40 is not computed

Intermediate fields

\(\Q(\sqrt{33}) \)

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

Sibling fields

Degree 24 sibling: data not computed
Arithmetically equvalently 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 R R $22$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $22$ $22$ ${\href{/LocalNumberField/29.11.0.1}{11} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.11.0.1}{11} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.4.8.3$x^{4} + 6 x^{2} + 4 x + 14$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.9.15$x^{6} + 6 x^{4} + 6 x^{3} + 12$$6$$1$$9$$S_3\times C_3$$[3/2, 2]_{2}$
3.6.9.15$x^{6} + 6 x^{4} + 6 x^{3} + 12$$6$$1$$9$$S_3\times C_3$$[3/2, 2]_{2}$
3.6.9.15$x^{6} + 6 x^{4} + 6 x^{3} + 12$$6$$1$$9$$S_3\times C_3$$[3/2, 2]_{2}$
11Data not computed