Properties

Label 22.0.178400853291024459894627.1
Degree 22
Signature $[0, 11]$
Discriminant $-\,3^{11}\cdot 1003532779^{2}$
Ramified primes $3, 1003532779$
Class number 1 (GRH)
Class group Trivial (GRH)
Galois Group 22T47

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, 4, 0, 1, 0, 10, 3, 5, 0, 13, -3, 7, -1, 9, -3, 6, -1, 5, 0, 2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 2*x^20 + 5*x^18 - x^17 + 6*x^16 - 3*x^15 + 9*x^14 - x^13 + 7*x^12 - 3*x^11 + 13*x^10 + 5*x^8 + 3*x^7 + 10*x^6 + x^4 + 4*x^2 + 2*x + 1)
gp: K = bnfinit(x^22 + 2*x^20 + 5*x^18 - x^17 + 6*x^16 - 3*x^15 + 9*x^14 - x^13 + 7*x^12 - 3*x^11 + 13*x^10 + 5*x^8 + 3*x^7 + 10*x^6 + x^4 + 4*x^2 + 2*x + 1, 1)

Normalized defining polynomial

\(x^{22} \) \(\mathstrut +\mathstrut 2 x^{20} \) \(\mathstrut +\mathstrut 5 x^{18} \) \(\mathstrut -\mathstrut x^{17} \) \(\mathstrut +\mathstrut 6 x^{16} \) \(\mathstrut -\mathstrut 3 x^{15} \) \(\mathstrut +\mathstrut 9 x^{14} \) \(\mathstrut -\mathstrut x^{13} \) \(\mathstrut +\mathstrut 7 x^{12} \) \(\mathstrut -\mathstrut 3 x^{11} \) \(\mathstrut +\mathstrut 13 x^{10} \) \(\mathstrut +\mathstrut 5 x^{8} \) \(\mathstrut +\mathstrut 3 x^{7} \) \(\mathstrut +\mathstrut 10 x^{6} \) \(\mathstrut +\mathstrut x^{4} \) \(\mathstrut +\mathstrut 4 x^{2} \) \(\mathstrut +\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-178400853291024459894627=-\,3^{11}\cdot 1003532779^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $3, 1003532779$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{15} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{20} + \frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{5397333} a^{21} + \frac{441266}{5397333} a^{20} - \frac{301661}{5397333} a^{19} - \frac{39386}{1799111} a^{18} - \frac{871243}{5397333} a^{17} + \frac{316840}{5397333} a^{16} - \frac{1794586}{5397333} a^{15} + \frac{1514548}{5397333} a^{14} - \frac{823615}{5397333} a^{13} + \frac{506099}{1799111} a^{12} + \frac{1898719}{5397333} a^{11} - \frac{152372}{1799111} a^{10} + \frac{393048}{1799111} a^{9} + \frac{820146}{1799111} a^{8} - \frac{1880657}{5397333} a^{7} - \frac{1257233}{5397333} a^{6} - \frac{309008}{5397333} a^{5} - \frac{101438}{5397333} a^{4} + \frac{2540284}{5397333} a^{3} - \frac{2345450}{5397333} a^{2} + \frac{1847941}{5397333} a - \frac{933665}{5397333}$

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:  \( -\frac{2810593}{5397333} a^{21} + \frac{1635334}{5397333} a^{20} - \frac{2117703}{1799111} a^{19} + \frac{3344648}{5397333} a^{18} - \frac{5175742}{1799111} a^{17} + \frac{10279327}{5397333} a^{16} - \frac{21606998}{5397333} a^{15} + \frac{17570141}{5397333} a^{14} - \frac{33587785}{5397333} a^{13} + \frac{5417000}{1799111} a^{12} - \frac{24871499}{5397333} a^{11} + \frac{16467355}{5397333} a^{10} - \frac{41454842}{5397333} a^{9} + \frac{5366617}{1799111} a^{8} - \frac{19521955}{5397333} a^{7} - \frac{4275268}{5397333} a^{6} - \frac{7771465}{1799111} a^{5} + \frac{3009008}{5397333} a^{4} - \frac{4389352}{5397333} a^{3} - \frac{1071251}{5397333} a^{2} - \frac{4088518}{1799111} a - \frac{135456}{1799111} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{135456}{1799111} a^{21} - \frac{2810593}{5397333} a^{20} + \frac{2448070}{5397333} a^{19} - \frac{2117703}{1799111} a^{18} + \frac{5376488}{5397333} a^{17} - \frac{5311198}{1799111} a^{16} + \frac{12717535}{5397333} a^{15} - \frac{22826102}{5397333} a^{14} + \frac{21227453}{5397333} a^{13} - \frac{33994153}{5397333} a^{12} + \frac{6365192}{1799111} a^{11} - \frac{26090603}{5397333} a^{10} + \frac{21750139}{5397333} a^{9} - \frac{41454842}{5397333} a^{8} + \frac{6043897}{1799111} a^{7} - \frac{18302851}{5397333} a^{6} - \frac{211588}{5397333} a^{5} - \frac{7771465}{1799111} a^{4} + \frac{3415376}{5397333} a^{3} - \frac{4389352}{5397333} a^{2} + \frac{554221}{5397333} a - \frac{3817606}{1799111} \),  \( \frac{13146215}{5397333} a^{21} + \frac{520521}{1799111} a^{20} + \frac{6881174}{1799111} a^{19} + \frac{1924477}{1799111} a^{18} + \frac{55202282}{5397333} a^{17} - \frac{351090}{1799111} a^{16} + \frac{51191879}{5397333} a^{15} - \frac{13287413}{5397333} a^{14} + \frac{81707636}{5397333} a^{13} + \frac{28405441}{5397333} a^{12} + \frac{13071900}{1799111} a^{11} - \frac{6827959}{5397333} a^{10} + \frac{44922209}{1799111} a^{9} + \frac{47069564}{5397333} a^{8} - \frac{2788421}{1799111} a^{7} + \frac{75481936}{5397333} a^{6} + \frac{116028835}{5397333} a^{5} + \frac{503584}{5397333} a^{4} - \frac{29132048}{5397333} a^{3} + \frac{29222654}{5397333} a^{2} + \frac{16084401}{1799111} a + \frac{29290429}{5397333} \),  \( \frac{2914040}{5397333} a^{21} - \frac{4633946}{5397333} a^{20} + \frac{7807048}{5397333} a^{19} - \frac{8672473}{5397333} a^{18} + \frac{5439087}{1799111} a^{17} - \frac{7643274}{1799111} a^{16} + \frac{27333146}{5397333} a^{15} - \frac{32097508}{5397333} a^{14} + \frac{13594317}{1799111} a^{13} - \frac{12320291}{1799111} a^{12} + \frac{9103563}{1799111} a^{11} - \frac{23109127}{5397333} a^{10} + \frac{14745544}{1799111} a^{9} - \frac{49184566}{5397333} a^{8} + \frac{28859927}{5397333} a^{7} + \frac{3228196}{5397333} a^{6} + \frac{193208}{1799111} a^{5} - \frac{7347184}{1799111} a^{4} + \frac{16392973}{5397333} a^{3} - \frac{4183661}{5397333} a^{2} + \frac{1494144}{1799111} a - \frac{961963}{5397333} \),  \( \frac{9092608}{5397333} a^{21} + \frac{1949965}{5397333} a^{20} + \frac{4740386}{1799111} a^{19} + \frac{5342473}{5397333} a^{18} + \frac{37452341}{5397333} a^{17} + \frac{3465530}{5397333} a^{16} + \frac{32565739}{5397333} a^{15} - \frac{1645737}{1799111} a^{14} + \frac{50570243}{5397333} a^{13} + \frac{27878062}{5397333} a^{12} + \frac{7407051}{1799111} a^{11} + \frac{532704}{1799111} a^{10} + \frac{84369250}{5397333} a^{9} + \frac{45462181}{5397333} a^{8} - \frac{2397031}{1799111} a^{7} + \frac{17748704}{1799111} a^{6} + \frac{26129573}{1799111} a^{5} + \frac{12466955}{5397333} a^{4} - \frac{25979498}{5397333} a^{3} + \frac{13953313}{5397333} a^{2} + \frac{34422725}{5397333} a + \frac{7474608}{1799111} \),  \( \frac{3310174}{5397333} a^{21} - \frac{395729}{5397333} a^{20} + \frac{4781983}{5397333} a^{19} + \frac{1392797}{5397333} a^{18} + \frac{12604940}{5397333} a^{17} - \frac{541541}{1799111} a^{16} + \frac{11732941}{5397333} a^{15} - \frac{497749}{1799111} a^{14} + \frac{18872615}{5397333} a^{13} + \frac{7076956}{5397333} a^{12} + \frac{1911459}{1799111} a^{11} + \frac{1611922}{5397333} a^{10} + \frac{12159592}{1799111} a^{9} + \frac{3149428}{5397333} a^{8} - \frac{3629230}{5397333} a^{7} + \frac{27384659}{5397333} a^{6} + \frac{26905435}{5397333} a^{5} - \frac{14741615}{5397333} a^{4} - \frac{94378}{5397333} a^{3} + \frac{4204219}{1799111} a^{2} + \frac{1821097}{1799111} a + \frac{675732}{1799111} \),  \( \frac{1516616}{5397333} a^{21} - \frac{1344345}{1799111} a^{20} + \frac{1543597}{1799111} a^{19} - \frac{7058617}{5397333} a^{18} + \frac{2934606}{1799111} a^{17} - \frac{5978220}{1799111} a^{16} + \frac{5690015}{1799111} a^{15} - \frac{7280783}{1799111} a^{14} + \frac{24076015}{5397333} a^{13} - \frac{26637058}{5397333} a^{12} + \frac{11927857}{5397333} a^{11} - \frac{10801382}{5397333} a^{10} + \frac{19309369}{5397333} a^{9} - \frac{12022578}{1799111} a^{8} + \frac{3771564}{1799111} a^{7} + \frac{10328491}{5397333} a^{6} - \frac{4681216}{1799111} a^{5} - \frac{16704197}{5397333} a^{4} + \frac{13666211}{5397333} a^{3} + \frac{3097781}{5397333} a^{2} - \frac{2245924}{5397333} a - \frac{2973980}{5397333} \),  \( \frac{1649137}{1799111} a^{21} - \frac{8779735}{5397333} a^{20} + \frac{4459830}{1799111} a^{19} - \frac{16239262}{5397333} a^{18} + \frac{29874142}{5397333} a^{17} - \frac{45704170}{5397333} a^{16} + \frac{17623607}{1799111} a^{15} - \frac{21250001}{1799111} a^{14} + \frac{82374466}{5397333} a^{13} - \frac{27577228}{1799111} a^{12} + \frac{61194208}{5397333} a^{11} - \frac{61797898}{5397333} a^{10} + \frac{97976017}{5397333} a^{9} - \frac{111138641}{5397333} a^{8} + \frac{55841230}{5397333} a^{7} - \frac{3502795}{5397333} a^{6} + \frac{17587739}{5397333} a^{5} - \frac{63628897}{5397333} a^{4} + \frac{40163119}{5397333} a^{3} - \frac{2600372}{5397333} a^{2} + \frac{6449605}{5397333} a - \frac{15994370}{5397333} \),  \( \frac{2905332}{1799111} a^{21} - \frac{3835979}{5397333} a^{20} + \frac{16710484}{5397333} a^{19} - \frac{6742082}{5397333} a^{18} + \frac{41554210}{5397333} a^{17} - \frac{26828362}{5397333} a^{16} + \frac{51944777}{5397333} a^{15} - \frac{15306752}{1799111} a^{14} + \frac{83318222}{5397333} a^{13} - \frac{13235967}{1799111} a^{12} + \frac{56099627}{5397333} a^{11} - \frac{48538483}{5397333} a^{10} + \frac{40178935}{1799111} a^{9} - \frac{16681773}{1799111} a^{8} + \frac{34565842}{5397333} a^{7} + \frac{14466397}{5397333} a^{6} + \frac{23339675}{1799111} a^{5} - \frac{42860404}{5397333} a^{4} + \frac{7428496}{5397333} a^{3} + \frac{85089}{1799111} a^{2} + \frac{31373185}{5397333} a + \frac{1228966}{5397333} \),  \( \frac{4747603}{5397333} a^{21} + \frac{4369891}{5397333} a^{20} + \frac{4848301}{5397333} a^{19} + \frac{3309138}{1799111} a^{18} + \frac{4940746}{1799111} a^{17} + \frac{19014974}{5397333} a^{16} + \frac{1518235}{5397333} a^{15} + \frac{21467185}{5397333} a^{14} + \frac{3396332}{5397333} a^{13} + \frac{17403532}{1799111} a^{12} - \frac{3826538}{1799111} a^{11} + \frac{26324132}{5397333} a^{10} + \frac{24389119}{5397333} a^{9} + \frac{71451971}{5397333} a^{8} - \frac{31521257}{5397333} a^{7} + \frac{14071549}{1799111} a^{6} + \frac{50350903}{5397333} a^{5} + \frac{27465142}{5397333} a^{4} - \frac{32073251}{5397333} a^{3} + \frac{15434614}{5397333} a^{2} + \frac{8131676}{1799111} a + \frac{6768786}{1799111} \),  \( \frac{6525079}{5397333} a^{21} + \frac{3662947}{5397333} a^{20} + \frac{7707550}{5397333} a^{19} + \frac{9739013}{5397333} a^{18} + \frac{22075040}{5397333} a^{17} + \frac{15196262}{5397333} a^{16} + \frac{3255803}{1799111} a^{15} + \frac{17831626}{5397333} a^{14} + \frac{14706179}{5397333} a^{13} + \frac{17341213}{1799111} a^{12} - \frac{3121535}{1799111} a^{11} + \frac{28114736}{5397333} a^{10} + \frac{13131849}{1799111} a^{9} + \frac{23990703}{1799111} a^{8} - \frac{42996104}{5397333} a^{7} + \frac{66878947}{5397333} a^{6} + \frac{53840729}{5397333} a^{5} + \frac{6055099}{1799111} a^{4} - \frac{35472985}{5397333} a^{3} + \frac{9304462}{1799111} a^{2} + \frac{16839691}{5397333} a + \frac{21529436}{5397333} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 622.138184002 \) (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

\(\Q(\sqrt{-3}) \), Deg 11

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.12.6.2$x^{12} + 18 x^{11} + 255 x^{10} - 378 x^{9} + 54 x^{8} - 459 x^{7} - 675 x^{6} + 486 x^{5} + 405 x^{4} - 729 x^{2} + 729$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
1003532779Data not computed