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Properties

Label 22.6.13651261303...7216.1
Degree $22$
Signature $[6, 8]$
Discriminant $2^{68}\cdot 3^{21}\cdot 11\cdot 31\cdot 37\cdot 337^{8}\cdot 310501^{8}\cdot 243830141$
Root discriminant $74{,}093.58$
Ramified primes $2, 3, 11, 31, 37, 337, 310501, 243830141$
Class number Not computed
Class group Not computed
Galois group 22T44

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-36916858667964, 0, -77942112687873, 0, -22774387512564, 0, -2056157959803, 0, 5363172192, 0, 9072108672, 0, 333658776, 0, -3803688, 0, -343692, 0, -2341, 0, 84, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 84*x^20 - 2341*x^18 - 343692*x^16 - 3803688*x^14 + 333658776*x^12 + 9072108672*x^10 + 5363172192*x^8 - 2056157959803*x^6 - 22774387512564*x^4 - 77942112687873*x^2 - 36916858667964)
 
gp: K = bnfinit(x^22 + 84*x^20 - 2341*x^18 - 343692*x^16 - 3803688*x^14 + 333658776*x^12 + 9072108672*x^10 + 5363172192*x^8 - 2056157959803*x^6 - 22774387512564*x^4 - 77942112687873*x^2 - 36916858667964, 1)
 

Normalized defining polynomial

\( x^{22} + 84 x^{20} - 2341 x^{18} - 343692 x^{16} - 3803688 x^{14} + 333658776 x^{12} + 9072108672 x^{10} + 5363172192 x^{8} - 2056157959803 x^{6} - 22774387512564 x^{4} - 77942112687873 x^{2} - 36916858667964 \)

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:  \(136512613038986913742008486695416726751511776495295634952214697836102284383214104319797066438605390496137216=2^{68}\cdot 3^{21}\cdot 11\cdot 31\cdot 37\cdot 337^{8}\cdot 310501^{8}\cdot 243830141\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74{,}093.58$
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:  $2, 3, 11, 31, 37, 337, 310501, 243830141$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{17} - \frac{1}{2} a^{16} + \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718491205}{57887320607168480035725746532474501690526662399435613406516} a^{18} - \frac{5260628338705088304708544940283463866244417696779481222833}{14471830151792120008931436633118625422631665599858903351629} a^{16} + \frac{1862815518614171432916916372115606405079794023195673116011}{14471830151792120008931436633118625422631665599858903351629} a^{14} - \frac{5971436442736102431805173807664951683990214041730634988600}{14471830151792120008931436633118625422631665599858903351629} a^{12} + \frac{4783913695076330561191782344476371722020399003380016029135}{14471830151792120008931436633118625422631665599858903351629} a^{10} + \frac{4129638676193580634336740984893486671296325384433486391522}{14471830151792120008931436633118625422631665599858903351629} a^{8} - \frac{1388090939202489857342431684589240974415729898068205871800}{14471830151792120008931436633118625422631665599858903351629} a^{6} + \frac{5365070608331306529419303327457602653555706182462125407541}{57887320607168480035725746532474501690526662399435613406516} a^{4} - \frac{27439428531960532971019149142947910324725200397562224966789}{57887320607168480035725746532474501690526662399435613406516} a^{2} - \frac{376412577946053724145161733249890854754310389639865058102}{1315620922890192728084676057556238674784696872714445759239}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{21} - \frac{2950442336895763005424408194123522436283571976756718491205}{57887320607168480035725746532474501690526662399435613406516} a^{19} + \frac{3950573474381943399514346752551697690142830206299940905963}{28943660303584240017862873266237250845263331199717806703258} a^{17} + \frac{1862815518614171432916916372115606405079794023195673116011}{14471830151792120008931436633118625422631665599858903351629} a^{15} - \frac{5971436442736102431805173807664951683990214041730634988600}{14471830151792120008931436633118625422631665599858903351629} a^{13} + \frac{4783913695076330561191782344476371722020399003380016029135}{14471830151792120008931436633118625422631665599858903351629} a^{11} + \frac{4129638676193580634336740984893486671296325384433486391522}{14471830151792120008931436633118625422631665599858903351629} a^{9} - \frac{1388090939202489857342431684589240974415729898068205871800}{14471830151792120008931436633118625422631665599858903351629} a^{7} + \frac{5365070608331306529419303327457602653555706182462125407541}{57887320607168480035725746532474501690526662399435613406516} a^{5} - \frac{27439428531960532971019149142947910324725200397562224966789}{57887320607168480035725746532474501690526662399435613406516} a^{3} + \frac{562795766998085279794352591056456965276076093434715643035}{2631241845780385456169352115112477349569393745428891518478} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T44:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 16220160
The 104 conjugacy class representatives for t22n44 are not computed
Character table for t22n44 is not computed

Intermediate fields

11.11.118769262421915560193703211428553337469927424.1

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 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 R R $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ R $22$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ R R ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
2Data not computed
3Data not computed
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
11.8.0.1$x^{8} + x^{2} - 2 x + 6$$1$$8$$0$$C_8$$[\ ]^{8}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$
$37$37.2.1.2$x^{2} + 74$$2$$1$$1$$C_2$$[\ ]_{2}$
37.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
37.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
37.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
37.5.0.1$x^{5} - x + 13$$1$$5$$0$$C_5$$[\ ]^{5}$
337Data not computed
310501Data not computed
243830141Data not computed