↑

Properties

Label 22.0.12904509791...0032.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{57}\cdot 11^{23}$
Root discriminant $73.90$
Ramified primes $2, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 22T41

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5643763, -9168896, 5021863, -7434240, 5775209, 315392, 2309285, 540672, 141086, 22528, 25894, -4096, 23474, 0, -726, 0, -33, 0, 99, 0, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 11*x^20 + 99*x^18 - 33*x^16 - 726*x^14 + 23474*x^12 - 4096*x^11 + 25894*x^10 + 22528*x^9 + 141086*x^8 + 540672*x^7 + 2309285*x^6 + 315392*x^5 + 5775209*x^4 - 7434240*x^3 + 5021863*x^2 - 9168896*x + 5643763)
 
gp: K = bnfinit(x^22 - 11*x^20 + 99*x^18 - 33*x^16 - 726*x^14 + 23474*x^12 - 4096*x^11 + 25894*x^10 + 22528*x^9 + 141086*x^8 + 540672*x^7 + 2309285*x^6 + 315392*x^5 + 5775209*x^4 - 7434240*x^3 + 5021863*x^2 - 9168896*x + 5643763, 1)
 

Normalized defining polynomial

\( x^{22} - 11 x^{20} + 99 x^{18} - 33 x^{16} - 726 x^{14} + 23474 x^{12} - 4096 x^{11} + 25894 x^{10} + 22528 x^{9} + 141086 x^{8} + 540672 x^{7} + 2309285 x^{6} + 315392 x^{5} + 5775209 x^{4} - 7434240 x^{3} + 5021863 x^{2} - 9168896 x + 5643763 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 11]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-129045097915537907803232946851577207980032=-\,2^{57}\cdot 11^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.90$
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:  $2, 11$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{16} a^{6} + \frac{1}{16} a^{4} - \frac{1}{16} a^{2} - \frac{1}{16}$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{1}{16} a + \frac{1}{8}$, $\frac{1}{32} a^{8} - \frac{1}{16} a^{4} + \frac{1}{32}$, $\frac{1}{64} a^{9} - \frac{1}{64} a^{8} - \frac{1}{32} a^{7} - \frac{1}{32} a^{6} - \frac{1}{16} a^{5} + \frac{1}{32} a^{3} + \frac{1}{32} a^{2} + \frac{3}{64} a + \frac{1}{64}$, $\frac{1}{128} a^{10} + \frac{1}{128} a^{8} - \frac{1}{64} a^{6} + \frac{7}{64} a^{4} + \frac{1}{128} a^{2} - \frac{15}{128}$, $\frac{1}{1408} a^{11} - \frac{1}{128} a^{9} + \frac{1}{64} a^{7} - \frac{3}{64} a^{5} - \frac{5}{128} a^{3} - \frac{1}{4} a^{2} + \frac{39}{128} a + \frac{13}{44}$, $\frac{1}{2816} a^{12} - \frac{1}{256} a^{8} - \frac{7}{256} a^{4} + \frac{3}{11} a + \frac{101}{256}$, $\frac{1}{5632} a^{13} - \frac{1}{5632} a^{12} - \frac{1}{2816} a^{11} - \frac{1}{256} a^{10} + \frac{1}{512} a^{9} + \frac{7}{512} a^{8} - \frac{1}{128} a^{7} - \frac{3}{128} a^{6} + \frac{5}{512} a^{5} - \frac{53}{512} a^{4} - \frac{27}{256} a^{3} - \frac{243}{2816} a^{2} - \frac{1219}{5632} a - \frac{645}{5632}$, $\frac{1}{11264} a^{14} - \frac{1}{11264} a^{12} + \frac{3}{1024} a^{10} - \frac{11}{1024} a^{8} + \frac{17}{1024} a^{6} + \frac{127}{1024} a^{4} + \frac{3}{44} a^{3} + \frac{73}{1024} a^{2} + \frac{19}{44} a - \frac{209}{1024}$, $\frac{1}{22528} a^{15} - \frac{1}{22528} a^{14} - \frac{1}{22528} a^{13} - \frac{3}{22528} a^{12} + \frac{1}{22528} a^{11} + \frac{5}{2048} a^{10} - \frac{11}{2048} a^{9} - \frac{9}{2048} a^{8} + \frac{17}{2048} a^{7} + \frac{31}{2048} a^{6} - \frac{65}{2048} a^{5} + \frac{2319}{22528} a^{4} + \frac{1091}{22528} a^{3} - \frac{2187}{22528} a^{2} - \frac{8123}{22528} a + \frac{10327}{22528}$, $\frac{1}{22528} a^{16} - \frac{1}{256} a^{10} - \frac{13}{1024} a^{8} + \frac{1}{128} a^{6} + \frac{3}{88} a^{5} - \frac{11}{256} a^{4} + \frac{63}{256} a^{2} - \frac{1}{8} a - \frac{305}{2048}$, $\frac{1}{495616} a^{17} - \frac{1}{495616} a^{16} - \frac{3}{247808} a^{15} - \frac{3}{247808} a^{14} - \frac{1}{247808} a^{13} + \frac{3}{247808} a^{12} - \frac{23}{247808} a^{11} + \frac{27}{22528} a^{10} + \frac{5}{704} a^{9} + \frac{119}{11264} a^{8} - \frac{443}{22528} a^{7} - \frac{7305}{247808} a^{6} + \frac{11265}{247808} a^{5} + \frac{5957}{247808} a^{4} - \frac{4317}{247808} a^{3} + \frac{16227}{247808} a^{2} - \frac{93633}{495616} a + \frac{83197}{495616}$, $\frac{1}{1982464} a^{18} + \frac{15}{1982464} a^{16} + \frac{1}{61952} a^{15} + \frac{5}{123904} a^{14} - \frac{1}{15488} a^{13} + \frac{17}{247808} a^{12} - \frac{9}{61952} a^{11} + \frac{29}{8192} a^{10} + \frac{9}{2816} a^{9} - \frac{1019}{90112} a^{8} + \frac{1311}{61952} a^{7} + \frac{277}{22528} a^{6} + \frac{105}{3872} a^{5} - \frac{6395}{123904} a^{4} - \frac{1303}{61952} a^{3} - \frac{463603}{1982464} a^{2} + \frac{6615}{30976} a - \frac{446805}{1982464}$, $\frac{1}{3964928} a^{19} - \frac{1}{3964928} a^{18} - \frac{1}{3964928} a^{17} + \frac{3}{360448} a^{16} - \frac{1}{123904} a^{15} + \frac{1}{61952} a^{14} - \frac{29}{495616} a^{13} - \frac{87}{495616} a^{12} + \frac{413}{1982464} a^{11} - \frac{199}{180224} a^{10} + \frac{973}{180224} a^{9} - \frac{20351}{1982464} a^{8} + \frac{9309}{495616} a^{7} - \frac{9457}{495616} a^{6} - \frac{173}{2816} a^{5} - \frac{12219}{123904} a^{4} + \frac{283133}{3964928} a^{3} + \frac{475235}{3964928} a^{2} - \frac{1418037}{3964928} a + \frac{975093}{3964928}$, $\frac{1}{55508992} a^{20} + \frac{3}{27754496} a^{19} - \frac{1}{6938624} a^{18} - \frac{1}{2523136} a^{17} - \frac{15}{7929856} a^{16} - \frac{9}{867328} a^{15} + \frac{151}{6938624} a^{14} - \frac{95}{3469312} a^{13} + \frac{347}{2523136} a^{12} - \frac{4905}{13877248} a^{11} - \frac{79}{57344} a^{10} - \frac{86587}{13877248} a^{9} - \frac{401253}{27754496} a^{8} - \frac{106017}{3469312} a^{7} - \frac{3545}{630784} a^{6} - \frac{2363}{108416} a^{5} + \frac{451867}{7929856} a^{4} - \frac{3284873}{27754496} a^{3} - \frac{9261}{247808} a^{2} + \frac{744971}{2523136} a - \frac{11498077}{55508992}$, $\frac{1}{26490693581406208} a^{21} + \frac{29332957}{26490693581406208} a^{20} + \frac{1379310087}{13245346790703104} a^{19} + \frac{108233505}{1204122435518464} a^{18} - \frac{9145271503}{26490693581406208} a^{17} - \frac{182926033075}{26490693581406208} a^{16} + \frac{72985534583}{3311336697675776} a^{15} - \frac{11763812079}{301030608879616} a^{14} + \frac{64293840679}{1892192398671872} a^{13} - \frac{1591522641235}{13245346790703104} a^{12} - \frac{32660940201}{946096199335936} a^{11} + \frac{12704392989469}{6622673395351552} a^{10} + \frac{68703102746933}{13245346790703104} a^{9} + \frac{31537558968001}{13245346790703104} a^{8} - \frac{82974342819}{3909488427008} a^{7} + \frac{24218622574535}{3311336697675776} a^{6} + \frac{822153941009917}{26490693581406208} a^{5} - \frac{2425974690206103}{26490693581406208} a^{4} + \frac{118366184568845}{1204122435518464} a^{3} + \frac{691414248872275}{13245346790703104} a^{2} - \frac{12028706584504251}{26490693581406208} a - \frac{4863114693803}{19521513324544}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $10$
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:  \( 1485021311930 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T41:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 887040
The 21 conjugacy class representatives for t22n41
Character table for t22n41 is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$[\ ]^{2}$
2.4.9.2$x^{4} - 2 x^{2} + 2$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.8.22.24$x^{8} + 8 x^{5} + 2 x^{4} + 60$$4$$2$$22$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 4]^{2}$
2.8.26.40$x^{8} + 4 x^{6} + 8 x^{5} + 4 x^{4} + 8 x^{3} + 6$$8$$1$$26$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 3, 7/2, 4]^{2}$
11Data not computed