Properties

Label 22.0.6293360649840402636051867.1
Degree 22
Signature $[0, 11]$
Discriminant $-\,3^{11}\cdot 64661^{2}\cdot 92179^{2}$
Ramified primes $3, 64661, 92179$
Class number 1 (GRH)
Class group Trivial (GRH)
Galois Group 22T47

Related objects

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Show commands for: Magma / SageMath / Pari/GP

magma: K<a> := NumberField(PolynomialRing(Rationals())![1, 1, 1, -2, -1, -2, 5, -1, -2, -4, 6, 1, -2, -3, 5, 0, -1, -3, 3, -1, 0, -1, 1]);
sage: K = NumberField(x^22 - x^21 - x^19 + 3*x^18 - 3*x^17 - x^16 + 5*x^14 - 3*x^13 - 2*x^12 + x^11 + 6*x^10 - 4*x^9 - 2*x^8 - x^7 + 5*x^6 - 2*x^5 - x^4 - 2*x^3 + x^2 + x + 1,"a")
gp: K = bnfinit(x^22 - x^21 - x^19 + 3*x^18 - 3*x^17 - x^16 + 5*x^14 - 3*x^13 - 2*x^12 + x^11 + 6*x^10 - 4*x^9 - 2*x^8 - x^7 + 5*x^6 - 2*x^5 - x^4 - 2*x^3 + x^2 + x + 1, 1)

Normalized defining polynomial

\(x^{22} \) \(\mathstrut -\mathstrut x^{21} \) \(\mathstrut -\mathstrut x^{19} \) \(\mathstrut +\mathstrut 3 x^{18} \) \(\mathstrut -\mathstrut 3 x^{17} \) \(\mathstrut -\mathstrut x^{16} \) \(\mathstrut +\mathstrut 5 x^{14} \) \(\mathstrut -\mathstrut 3 x^{13} \) \(\mathstrut -\mathstrut 2 x^{12} \) \(\mathstrut +\mathstrut x^{11} \) \(\mathstrut +\mathstrut 6 x^{10} \) \(\mathstrut -\mathstrut 4 x^{9} \) \(\mathstrut -\mathstrut 2 x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut +\mathstrut 5 x^{6} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut -\mathstrut x^{4} \) \(\mathstrut -\mathstrut 2 x^{3} \) \(\mathstrut +\mathstrut x^{2} \) \(\mathstrut +\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-6293360649840402636051867=-\,3^{11}\cdot 64661^{2}\cdot 92179^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $3, 64661, 92179$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{5981} a^{21} - \frac{1485}{5981} a^{20} + \frac{2732}{5981} a^{19} + \frac{829}{5981} a^{18} + \frac{1853}{5981} a^{17} + \frac{1405}{5981} a^{16} + \frac{2348}{5981} a^{15} + \frac{2491}{5981} a^{14} - \frac{381}{5981} a^{13} - \frac{2794}{5981} a^{12} + \frac{1461}{5981} a^{11} + \frac{2980}{5981} a^{10} - \frac{2355}{5981} a^{9} + \frac{1912}{5981} a^{8} - \frac{2416}{5981} a^{7} + \frac{2724}{5981} a^{6} + \frac{745}{5981} a^{5} + \frac{903}{5981} a^{4} - \frac{309}{5981} a^{3} - \frac{1983}{5981} a^{2} + \frac{121}{5981} a - \frac{133}{5981}$

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

Class group and class number

Trivial Abelian group, 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:  $10$
magma: UnitRank(K);
sage: UK.rank()
gp: #K.fu
Torsion generator:  \( \frac{975}{5981} a^{21} - \frac{473}{5981} a^{20} + \frac{2155}{5981} a^{19} - \frac{5141}{5981} a^{18} + \frac{6394}{5981} a^{17} - \frac{5755}{5981} a^{16} + \frac{4558}{5981} a^{15} - \frac{11523}{5981} a^{14} + \frac{11309}{5981} a^{13} - \frac{2795}{5981} a^{12} + \frac{997}{5981} a^{11} - \frac{7247}{5981} a^{10} + \frac{12541}{5981} a^{9} - \frac{1872}{5981} a^{8} + \frac{914}{5981} a^{7} - \frac{11626}{5981} a^{6} + \frac{14636}{5981} a^{5} - \frac{4763}{5981} a^{4} + \frac{3756}{5981} a^{3} - \frac{7543}{5981} a^{2} + \frac{4336}{5981} a + \frac{1907}{5981} \)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{502}{5981} a^{21} + \frac{2155}{5981} a^{20} - \frac{4166}{5981} a^{19} + \frac{3469}{5981} a^{18} - \frac{2830}{5981} a^{17} + \frac{5533}{5981} a^{16} - \frac{11523}{5981} a^{15} + \frac{6434}{5981} a^{14} + \frac{130}{5981} a^{13} + \frac{2947}{5981} a^{12} - \frac{8222}{5981} a^{11} + \frac{6691}{5981} a^{10} + \frac{2028}{5981} a^{9} + \frac{2864}{5981} a^{8} - \frac{10651}{5981} a^{7} + \frac{9761}{5981} a^{6} - \frac{2813}{5981} a^{5} + \frac{4731}{5981} a^{4} - \frac{5593}{5981} a^{3} + \frac{3361}{5981} a^{2} + \frac{932}{5981} a - \frac{975}{5981} \),  \( \frac{2234}{5981} a^{21} - \frac{4016}{5981} a^{20} + \frac{2668}{5981} a^{19} - \frac{2124}{5981} a^{18} + \frac{6731}{5981} a^{17} - \frac{13217}{5981} a^{16} + \frac{6076}{5981} a^{15} + \frac{2564}{5981} a^{14} + \frac{4129}{5981} a^{13} - \frac{15575}{5981} a^{12} + \frac{10210}{5981} a^{11} + \frac{6448}{5981} a^{10} + \frac{2210}{5981} a^{9} - \frac{16969}{5981} a^{8} + \frac{15461}{5981} a^{7} - \frac{3242}{5981} a^{6} + \frac{1612}{5981} a^{5} - \frac{10257}{5981} a^{4} + \frac{9471}{5981} a^{3} - \frac{4082}{5981} a^{2} + \frac{1169}{5981} a + \frac{1928}{5981} \),  \( \frac{4899}{5981} a^{21} - \frac{8100}{5981} a^{20} + \frac{4571}{5981} a^{19} - \frac{5809}{5981} a^{18} + \frac{16632}{5981} a^{17} - \frac{24960}{5981} a^{16} + \frac{7370}{5981} a^{15} + \frac{2169}{5981} a^{14} + \frac{17496}{5981} a^{13} - \frac{27202}{5981} a^{12} + \frac{4163}{5981} a^{11} + \frac{11361}{5981} a^{10} + \frac{18147}{5981} a^{9} - \frac{35244}{5981} a^{8} + \frac{12377}{5981} a^{7} + \frac{1265}{5981} a^{6} + \frac{19288}{5981} a^{5} - \frac{26067}{5981} a^{4} + \frac{5383}{5981} a^{3} - \frac{1573}{5981} a^{2} + \frac{660}{5981} a + \frac{362}{5981} \),  \( \frac{4857}{5981} a^{21} - \frac{5540}{5981} a^{20} - \frac{2515}{5981} a^{19} + \frac{1240}{5981} a^{18} + \frac{10578}{5981} a^{17} - \frac{12198}{5981} a^{16} - \frac{13493}{5981} a^{15} + \frac{17167}{5981} a^{14} + \frac{15555}{5981} a^{13} - \frac{17512}{5981} a^{12} - \frac{15332}{5981} a^{11} + \frac{23764}{5981} a^{10} + \frac{21361}{5981} a^{9} - \frac{25833}{5981} a^{8} - \frac{17733}{5981} a^{7} + \frac{18439}{5981} a^{6} + \frac{11922}{5981} a^{5} - \frac{10164}{5981} a^{4} - \frac{17525}{5981} a^{3} + \frac{3960}{5981} a^{2} + \frac{1559}{5981} a + \frac{5948}{5981} \),  \( \frac{270}{5981} a^{21} - \frac{223}{5981} a^{20} + \frac{1977}{5981} a^{19} - \frac{3448}{5981} a^{18} + \frac{3887}{5981} a^{17} - \frac{3434}{5981} a^{16} + \frac{5955}{5981} a^{15} - \frac{9264}{5981} a^{14} + \frac{4788}{5981} a^{13} - \frac{774}{5981} a^{12} + \frac{5705}{5981} a^{11} - \frac{8816}{5981} a^{10} + \frac{4117}{5981} a^{9} + \frac{1874}{5981} a^{8} + \frac{5590}{5981} a^{7} - \frac{12145}{5981} a^{6} + \frac{3777}{5981} a^{5} - \frac{1411}{5981} a^{4} + \frac{6285}{5981} a^{3} - \frac{9082}{5981} a^{2} + \frac{2765}{5981} a - \frac{24}{5981} \),  \( \frac{623}{5981} a^{21} - \frac{4081}{5981} a^{20} + \frac{3432}{5981} a^{19} + \frac{2101}{5981} a^{18} + \frac{86}{5981} a^{17} - \frac{9873}{5981} a^{16} + \frac{9421}{5981} a^{15} + \frac{8795}{5981} a^{14} - \frac{10085}{5981} a^{13} - \frac{18134}{5981} a^{12} + \frac{19034}{5981} a^{11} + \frac{8411}{5981} a^{10} - \frac{13782}{5981} a^{9} - \frac{16986}{5981} a^{8} + \frac{25968}{5981} a^{7} + \frac{10410}{5981} a^{6} - \frac{14345}{5981} a^{5} - \frac{11607}{5981} a^{4} + \frac{10847}{5981} a^{3} + \frac{2658}{5981} a^{2} - \frac{2370}{5981} a - \frac{5106}{5981} \),  \( \frac{1858}{5981} a^{21} - \frac{1889}{5981} a^{20} - \frac{1813}{5981} a^{19} + \frac{3165}{5981} a^{18} + \frac{3799}{5981} a^{17} - \frac{9188}{5981} a^{16} - \frac{3546}{5981} a^{15} + \frac{10946}{5981} a^{14} + \frac{3841}{5981} a^{13} - \frac{17687}{5981} a^{12} - \frac{836}{5981} a^{11} + \frac{22358}{5981} a^{10} - \frac{3479}{5981} a^{9} - \frac{18161}{5981} a^{8} + \frac{2803}{5981} a^{7} + \frac{19209}{5981} a^{6} - \frac{3382}{5981} a^{5} - \frac{14849}{5981} a^{4} + \frac{6035}{5981} a^{3} + \frac{5863}{5981} a^{2} - \frac{2460}{5981} a - \frac{1893}{5981} \),  \( \frac{1431}{5981} a^{21} - \frac{1780}{5981} a^{20} + \frac{3899}{5981} a^{19} - \frac{3920}{5981} a^{18} + \frac{8041}{5981} a^{17} - \frac{11023}{5981} a^{16} + \frac{10628}{5981} a^{15} - \frac{12017}{5981} a^{14} + \frac{11022}{5981} a^{13} - \frac{14868}{5981} a^{12} + \frac{9303}{5981} a^{11} - \frac{6054}{5981} a^{10} + \frac{9260}{5981} a^{9} - \frac{9207}{5981} a^{8} + \frac{11684}{5981} a^{7} - \frac{7549}{5981} a^{6} + \frac{13439}{5981} a^{5} - \frac{17646}{5981} a^{4} + \frac{12377}{5981} a^{3} - \frac{8660}{5981} a^{2} + \frac{5683}{5981} a - \frac{4912}{5981} \),  \( \frac{1808}{5981} a^{21} + \frac{589}{5981} a^{20} - \frac{850}{5981} a^{19} - \frac{2399}{5981} a^{18} + \frac{864}{5981} a^{17} + \frac{4296}{5981} a^{16} - \frac{7307}{5981} a^{15} - \frac{5946}{5981} a^{14} + \frac{4948}{5981} a^{13} + \frac{14355}{5981} a^{12} - \frac{8095}{5981} a^{11} - \frac{13003}{5981} a^{10} + \frac{12594}{5981} a^{9} + \frac{17821}{5981} a^{8} - \frac{13960}{5981} a^{7} - \frac{15314}{5981} a^{6} + \frac{7216}{5981} a^{5} + \frac{17754}{5981} a^{4} - \frac{8420}{5981} a^{3} - \frac{8626}{5981} a^{2} - \frac{2529}{5981} a + \frac{4757}{5981} \),  \( \frac{3992}{5981} a^{21} - \frac{6930}{5981} a^{20} + \frac{8762}{5981} a^{19} - \frac{10087}{5981} a^{18} + \frac{16622}{5981} a^{17} - \frac{25342}{5981} a^{16} + \frac{18932}{5981} a^{15} - \frac{14293}{5981} a^{14} + \frac{16165}{5981} a^{13} - \frac{23007}{5981} a^{12} + \frac{18780}{5981} a^{11} - \frac{6030}{5981} a^{10} + \frac{12934}{5981} a^{9} - \frac{22976}{5981} a^{8} + \frac{26605}{5981} a^{7} - \frac{17193}{5981} a^{6} + \frac{19426}{5981} a^{5} - \frac{19710}{5981} a^{4} + \frac{22482}{5981} a^{3} - \frac{15235}{5981} a^{2} + \frac{4552}{5981} a - \frac{4608}{5981} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 4402.38521796 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

22T47:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 79833600
Conjugacy class representatives for 22T47
Character table for 22T47

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.1.5960386319.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $22$ R ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ $22$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
64661Data not computed
92179Data not computed