Properties

Label 22.0.492568978448025181133471744.1
Degree 22
Signature $[0, 11]$
Discriminant $-\,2^{18}\cdot 11^{5}\cdot 19^{4}\cdot 547^{4}$
Ramified primes $2, 11, 19, 547$
Class number 1 (GRH)
Class group Trivial (GRH)
Galois Group 22T52

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 0, 77, 0, 66, 0, -158, 0, -195, 0, 72, 0, 153, 0, 29, 0, -22, 0, -5, 0, 2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 2*x^20 - 5*x^18 - 22*x^16 + 29*x^14 + 153*x^12 + 72*x^10 - 195*x^8 - 158*x^6 + 66*x^4 + 77*x^2 + 11)
gp: K = bnfinit(x^22 + 2*x^20 - 5*x^18 - 22*x^16 + 29*x^14 + 153*x^12 + 72*x^10 - 195*x^8 - 158*x^6 + 66*x^4 + 77*x^2 + 11, 1)

Normalized defining polynomial

\(x^{22} \) \(\mathstrut +\mathstrut 2 x^{20} \) \(\mathstrut -\mathstrut 5 x^{18} \) \(\mathstrut -\mathstrut 22 x^{16} \) \(\mathstrut +\mathstrut 29 x^{14} \) \(\mathstrut +\mathstrut 153 x^{12} \) \(\mathstrut +\mathstrut 72 x^{10} \) \(\mathstrut -\mathstrut 195 x^{8} \) \(\mathstrut -\mathstrut 158 x^{6} \) \(\mathstrut +\mathstrut 66 x^{4} \) \(\mathstrut +\mathstrut 77 x^{2} \) \(\mathstrut +\mathstrut 11 \)

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:  \(-492568978448025181133471744=-\,2^{18}\cdot 11^{5}\cdot 19^{4}\cdot 547^{4}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $2, 11, 19, 547$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{18} - \frac{1}{4} a^{17} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{5528348828} a^{20} - \frac{272817717}{1382087207} a^{18} - \frac{1}{4} a^{17} - \frac{42819692}{1382087207} a^{16} - \frac{1}{4} a^{15} - \frac{123044705}{1382087207} a^{14} - \frac{250642240}{1382087207} a^{12} - \frac{1}{4} a^{11} - \frac{28171175}{1382087207} a^{10} - \frac{1}{4} a^{9} - \frac{392133315}{2764174414} a^{8} - \frac{439687842}{1382087207} a^{6} + \frac{1}{4} a^{5} - \frac{287650587}{5528348828} a^{4} - \frac{1}{4} a^{3} + \frac{359620784}{1382087207} a^{2} + \frac{1}{4} a + \frac{389481833}{5528348828}$, $\frac{1}{5528348828} a^{21} + \frac{290816339}{5528348828} a^{19} + \frac{1210808439}{5528348828} a^{17} - \frac{1}{4} a^{16} - \frac{123044705}{1382087207} a^{15} - \frac{1}{4} a^{14} + \frac{379518247}{5528348828} a^{13} + \frac{1269402507}{5528348828} a^{11} + \frac{1}{4} a^{10} + \frac{494976946}{1382087207} a^{9} + \frac{1}{4} a^{8} - \frac{376664161}{5528348828} a^{7} + \frac{273609155}{1382087207} a^{5} - \frac{1}{4} a^{4} - \frac{2707778485}{5528348828} a^{3} - \frac{1}{4} a^{2} + \frac{389481833}{5528348828} a + \frac{1}{4}$

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:  \( -1 \)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{792725095}{2764174414} a^{20} + \frac{845642949}{2764174414} a^{18} - \frac{4822819285}{2764174414} a^{16} - \frac{6523716598}{1382087207} a^{14} + \frac{35529305711}{2764174414} a^{12} + \frac{89448549059}{2764174414} a^{10} - \frac{14444533207}{1382087207} a^{8} - \frac{136607873999}{2764174414} a^{6} - \frac{122842295}{1382087207} a^{4} + \frac{62883390701}{2764174414} a^{2} + \frac{7405295673}{2764174414} \),  \( \frac{130212107}{1382087207} a^{20} + \frac{351159339}{2764174414} a^{18} - \frac{767186051}{1382087207} a^{16} - \frac{4778588097}{2764174414} a^{14} + \frac{10662317313}{2764174414} a^{12} + \frac{16619452600}{1382087207} a^{10} - \frac{2366231549}{2764174414} a^{8} - \frac{51700747649}{2764174414} a^{6} - \frac{5722998043}{1382087207} a^{4} + \frac{13661929029}{1382087207} a^{2} + \frac{8786648363}{2764174414} \),  \( \frac{82067133}{1382087207} a^{21} - \frac{90652948}{1382087207} a^{20} + \frac{226279087}{2764174414} a^{19} - \frac{131116289}{2764174414} a^{18} - \frac{1015796519}{2764174414} a^{17} + \frac{584425643}{1382087207} a^{16} - \frac{1507241424}{1382087207} a^{15} + \frac{1289707498}{1382087207} a^{14} + \frac{3482615631}{1382087207} a^{13} - \frac{4565247214}{1382087207} a^{12} + \frac{10765147189}{1382087207} a^{11} - \frac{17546434451}{2764174414} a^{10} - \frac{2089045921}{1382087207} a^{9} + \frac{6860215196}{1382087207} a^{8} - \frac{17113364174}{1382087207} a^{7} + \frac{13801524156}{1382087207} a^{6} - \frac{1351764775}{2764174414} a^{5} - \frac{10668085715}{2764174414} a^{4} + \frac{18187624031}{2764174414} a^{3} - \frac{13528845471}{2764174414} a^{2} + \frac{783669952}{1382087207} a + \frac{740922315}{2764174414} \),  \( \frac{621954672}{1382087207} a^{21} - \frac{82067133}{1382087207} a^{20} + \frac{1551404837}{2764174414} a^{19} - \frac{226279087}{2764174414} a^{18} - \frac{3677106611}{1382087207} a^{17} + \frac{1015796519}{2764174414} a^{16} - \frac{21863451503}{2764174414} a^{15} + \frac{1507241424}{1382087207} a^{14} + \frac{26093579181}{1382087207} a^{13} - \frac{3482615631}{1382087207} a^{12} + \frac{150788251185}{2764174414} a^{11} - \frac{10765147189}{1382087207} a^{10} - \frac{10524351043}{1382087207} a^{9} + \frac{2089045921}{1382087207} a^{8} - \frac{112389138020}{1382087207} a^{7} + \frac{17113364174}{1382087207} a^{6} - \frac{38202455867}{2764174414} a^{5} + \frac{1351764775}{2764174414} a^{4} + \frac{99849352315}{2764174414} a^{3} - \frac{18187624031}{2764174414} a^{2} + \frac{13555162035}{1382087207} a - \frac{783669952}{1382087207} \),  \( \frac{27809283}{5528348828} a^{21} - \frac{1519432397}{5528348828} a^{20} + \frac{13537877}{5528348828} a^{19} - \frac{449715975}{1382087207} a^{18} - \frac{35202689}{2764174414} a^{17} + \frac{8986223935}{5528348828} a^{16} - \frac{162222587}{5528348828} a^{15} + \frac{25904070557}{5528348828} a^{14} + \frac{1049568311}{5528348828} a^{13} - \frac{16233212638}{1382087207} a^{12} + \frac{122535089}{2764174414} a^{11} - \frac{177755223951}{5528348828} a^{10} + \frac{446478131}{5528348828} a^{9} + \frac{34174578911}{5528348828} a^{8} + \frac{9588286087}{5528348828} a^{7} + \frac{64135777840}{1382087207} a^{6} + \frac{12145524109}{5528348828} a^{5} + \frac{9722562645}{2764174414} a^{4} - \frac{2773659603}{2764174414} a^{3} - \frac{112801721585}{5528348828} a^{2} - \frac{3260679123}{2764174414} a - \frac{9891847765}{2764174414} \),  \( \frac{266134191}{5528348828} a^{21} + \frac{261542197}{5528348828} a^{20} + \frac{674345561}{5528348828} a^{19} + \frac{87840017}{1382087207} a^{18} - \frac{641237135}{2764174414} a^{17} - \frac{1484814403}{5528348828} a^{16} - \frac{6686484679}{5528348828} a^{15} - \frac{4716759661}{5528348828} a^{14} + \frac{6105464775}{5528348828} a^{13} + \frac{5183226471}{2764174414} a^{12} + \frac{23974784903}{2764174414} a^{11} + \frac{32290950357}{5528348828} a^{10} + \frac{28606710819}{5528348828} a^{9} - \frac{237743625}{5528348828} a^{8} - \frac{61771696103}{5528348828} a^{7} - \frac{11228544917}{1382087207} a^{6} - \frac{50415096753}{5528348828} a^{5} - \frac{5857307863}{2764174414} a^{4} + \frac{13686273363}{2764174414} a^{3} + \frac{17902842683}{5528348828} a^{2} + \frac{4760465919}{1382087207} a + \frac{1229656891}{2764174414} \),  \( \frac{1534733111}{5528348828} a^{21} + \frac{302451419}{5528348828} a^{20} + \frac{1943220303}{5528348828} a^{19} + \frac{67717701}{1382087207} a^{18} - \frac{2231851955}{1382087207} a^{17} - \frac{1784334025}{5528348828} a^{16} - \frac{26898164991}{5528348828} a^{15} - \frac{4562622843}{5528348828} a^{14} + \frac{63493844095}{5528348828} a^{13} + \frac{3462716722}{1382087207} a^{12} + \frac{46213422775}{1382087207} a^{11} + \frac{30194618897}{5528348828} a^{10} - \frac{20264675033}{5528348828} a^{9} - \frac{12479272523}{5528348828} a^{8} - \frac{262270403113}{5528348828} a^{7} - \frac{10098884160}{1382087207} a^{6} - \frac{37301012185}{5528348828} a^{5} + \frac{2266703663}{1382087207} a^{4} + \frac{57154517445}{2764174414} a^{3} + \frac{18835903649}{5528348828} a^{2} + \frac{6371911146}{1382087207} a - \frac{548029862}{1382087207} \),  \( \frac{505236041}{1382087207} a^{21} + \frac{597613211}{2764174414} a^{20} + \frac{2619299201}{5528348828} a^{19} + \frac{1400341497}{5528348828} a^{18} - \frac{11731800113}{5528348828} a^{17} - \frac{1735255708}{1382087207} a^{16} - \frac{8954908037}{1382087207} a^{15} - \frac{20124738905}{5528348828} a^{14} + \frac{82752009339}{5528348828} a^{13} + \frac{50487852617}{5528348828} a^{12} + \frac{246669753131}{5528348828} a^{11} + \frac{34201379778}{1382087207} a^{10} - \frac{10681018665}{2764174414} a^{9} - \frac{22539322209}{5528348828} a^{8} - \frac{349897878913}{5528348828} a^{7} - \frac{185635553667}{5528348828} a^{6} - \frac{61471116611}{5528348828} a^{5} - \frac{8034572391}{2764174414} a^{4} + \frac{148790390151}{5528348828} a^{3} + \frac{34224800965}{2764174414} a^{2} + \frac{8235314666}{1382087207} a + \frac{9672062269}{5528348828} \),  \( \frac{2385920}{19466017} a^{21} + \frac{1022360195}{5528348828} a^{20} + \frac{12239317}{77864068} a^{19} + \frac{1522753979}{5528348828} a^{18} - \frac{28620319}{38932034} a^{17} - \frac{2907160191}{2764174414} a^{16} - \frac{170462477}{77864068} a^{15} - \frac{19472737785}{5528348828} a^{14} + \frac{402421187}{77864068} a^{13} + \frac{39143793413}{5528348828} a^{12} + \frac{297235319}{19466017} a^{11} + \frac{33848084895}{1382087207} a^{10} - \frac{183503801}{77864068} a^{9} + \frac{8080078483}{5528348828} a^{8} - \frac{1845561793}{77864068} a^{7} - \frac{196756262851}{5528348828} a^{6} - \frac{68427910}{19466017} a^{5} - \frac{64917983493}{5528348828} a^{4} + \frac{485153443}{38932034} a^{3} + \frac{45318192341}{2764174414} a^{2} + \frac{249210469}{77864068} a + \frac{9064802172}{1382087207} \),  \( \frac{254714171}{2764174414} a^{20} + \frac{183315641}{2764174414} a^{18} - \frac{769284451}{1382087207} a^{16} - \frac{3556982363}{2764174414} a^{14} + \frac{12284360861}{2764174414} a^{12} + \frac{11600628190}{1382087207} a^{10} - \frac{14747669305}{2764174414} a^{8} - \frac{30590836639}{2764174414} a^{6} + \frac{13337950355}{2764174414} a^{4} + \frac{4866083003}{1382087207} a^{2} - \frac{2530486308}{1382087207} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 19198.3237336 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

22T52:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 40874803200
Conjugacy class representatives for 22T52
Character table for 22T52

Intermediate fields

Deg 11

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ $22$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ R ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.6.5.1$x^{6} - 11$$6$$1$$5$$S_3\times C_2$$[\ ]_{6}^{2}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
19.14.0.1$x^{14} + x^{2} - x + 15$$1$$14$$0$$C_{14}$$[\ ]^{14}$
547Data not computed