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Properties

Label 22.20.2360782888...1680.1
Degree $22$
Signature $[20, 1]$
Discriminant $-\,2^{70}\cdot 3^{21}\cdot 5\cdot 11\cdot 29\cdot 43\cdot 337^{8}\cdot 310501^{8}\cdot 193926709$
Root discriminant $84{,}342.80$
Ramified primes $2, 3, 5, 11, 29, 43, 337, 310501, 193926709$
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![638422240164720, 0, -3355033131829681, 0, 697620189720184, 0, -47964749150331, 0, 150453098976, 0, 161768261952, 0, -10449620808, 0, 332234520, 0, -6153936, 0, 67179, 0, -400, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 400*x^20 + 67179*x^18 - 6153936*x^16 + 332234520*x^14 - 10449620808*x^12 + 161768261952*x^10 + 150453098976*x^8 - 47964749150331*x^6 + 697620189720184*x^4 - 3355033131829681*x^2 + 638422240164720)
 
gp: K = bnfinit(x^22 - 400*x^20 + 67179*x^18 - 6153936*x^16 + 332234520*x^14 - 10449620808*x^12 + 161768261952*x^10 + 150453098976*x^8 - 47964749150331*x^6 + 697620189720184*x^4 - 3355033131829681*x^2 + 638422240164720, 1)
 

Normalized defining polynomial

\( x^{22} - 400 x^{20} + 67179 x^{18} - 6153936 x^{16} + 332234520 x^{14} - 10449620808 x^{12} + 161768261952 x^{10} + 150453098976 x^{8} - 47964749150331 x^{6} + 697620189720184 x^{4} - 3355033131829681 x^{2} + 638422240164720 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2360782888136677539846811675588590672634700789304705468856315278608937941898270750476788224860384348791111680=-\,2^{70}\cdot 3^{21}\cdot 5\cdot 11\cdot 29\cdot 43\cdot 337^{8}\cdot 310501^{8}\cdot 193926709\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84{,}342.80$
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:  $2, 3, 5, 11, 29, 43, 337, 310501, 193926709$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{17} + \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{20} - \frac{2950442336895763005424408194123522436283571976756718491645}{57887320607168480035725746532474501690526662399435613406516} a^{18} - \frac{2603440021866950946320866384427251126804103995042417604338}{14471830151792120008931436633118625422631665599858903351629} a^{16} - \frac{3356847592750441243324607258186595391020285802604853780517}{14471830151792120008931436633118625422631665599858903351629} a^{14} + \frac{4163615942939049138070709307651308553005686994909891076697}{14471830151792120008931436633118625422631665599858903351629} a^{12} - \frac{3320541285504482433560338830940529710410172444674416705262}{14471830151792120008931436633118625422631665599858903351629} a^{10} + \frac{6252926343353724827564831908809365656199123199966382483329}{14471830151792120008931436633118625422631665599858903351629} a^{8} + \frac{4830212119368684812064988315978933499899662118653069122910}{14471830151792120008931436633118625422631665599858903351629} a^{6} + \frac{12221361899331131538812917411456345585141244491789918076509}{57887320607168480035725746532474501690526662399435613406516} a^{4} - \frac{18176715180622911668603038220634977692713965299259310434193}{57887320607168480035725746532474501690526662399435613406516} a^{2} + \frac{425080577449944343247954747025853065021723310081986501105}{1315620922890192728084676057556238674784696872714445759239}$, $\frac{1}{57887320607168480035725746532474501690526662399435613406516} a^{21} - \frac{2950442336895763005424408194123522436283571976756718491645}{57887320607168480035725746532474501690526662399435613406516} a^{19} - \frac{2603440021866950946320866384427251126804103995042417604338}{14471830151792120008931436633118625422631665599858903351629} a^{17} - \frac{3356847592750441243324607258186595391020285802604853780517}{14471830151792120008931436633118625422631665599858903351629} a^{15} + \frac{4163615942939049138070709307651308553005686994909891076697}{14471830151792120008931436633118625422631665599858903351629} a^{13} - \frac{3320541285504482433560338830940529710410172444674416705262}{14471830151792120008931436633118625422631665599858903351629} a^{11} + \frac{6252926343353724827564831908809365656199123199966382483329}{14471830151792120008931436633118625422631665599858903351629} a^{9} + \frac{4830212119368684812064988315978933499899662118653069122910}{14471830151792120008931436633118625422631665599858903351629} a^{7} + \frac{12221361899331131538812917411456345585141244491789918076509}{57887320607168480035725746532474501690526662399435613406516} a^{5} - \frac{18176715180622911668603038220634977692713965299259310434193}{57887320607168480035725746532474501690526662399435613406516} a^{3} + \frac{425080577449944343247954747025853065021723310081986501105}{1315620922890192728084676057556238674784696872714445759239} a$

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:  $20$
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 R $22$ R $22$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ R $16{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
3Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$11$$\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$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$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}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
29.8.0.1$x^{8} + x^{2} - 3 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
43Data not computed
337Data not computed
310501Data not computed
193926709Data not computed