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Properties

Label 22.0.70739409751...8051.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,13^{11}\cdot 23^{21}$
Root discriminant $71.91$
Ramified primes $13, 23$
Class number $138344$ (GRH)
Class group $[138344]$ (GRH)
Galois group $C_{22}$ (as 22T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![214677493, -210603112, 210603112, -180724318, 180724318, -115986931, 115986931, -51249544, 51249544, -15284329, 15284329, -3077953, 3077953, -417589, 417589, -37537, 37537, -2140, 2140, -70, 70, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 70*x^20 - 70*x^19 + 2140*x^18 - 2140*x^17 + 37537*x^16 - 37537*x^15 + 417589*x^14 - 417589*x^13 + 3077953*x^12 - 3077953*x^11 + 15284329*x^10 - 15284329*x^9 + 51249544*x^8 - 51249544*x^7 + 115986931*x^6 - 115986931*x^5 + 180724318*x^4 - 180724318*x^3 + 210603112*x^2 - 210603112*x + 214677493)
 
gp: K = bnfinit(x^22 - x^21 + 70*x^20 - 70*x^19 + 2140*x^18 - 2140*x^17 + 37537*x^16 - 37537*x^15 + 417589*x^14 - 417589*x^13 + 3077953*x^12 - 3077953*x^11 + 15284329*x^10 - 15284329*x^9 + 51249544*x^8 - 51249544*x^7 + 115986931*x^6 - 115986931*x^5 + 180724318*x^4 - 180724318*x^3 + 210603112*x^2 - 210603112*x + 214677493, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 70 x^{20} - 70 x^{19} + 2140 x^{18} - 2140 x^{17} + 37537 x^{16} - 37537 x^{15} + 417589 x^{14} - 417589 x^{13} + 3077953 x^{12} - 3077953 x^{11} + 15284329 x^{10} - 15284329 x^{9} + 51249544 x^{8} - 51249544 x^{7} + 115986931 x^{6} - 115986931 x^{5} + 180724318 x^{4} - 180724318 x^{3} + 210603112 x^{2} - 210603112 x + 214677493 \)

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:  \(-70739409751010214154180397092922226278051=-\,13^{11}\cdot 23^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.91$
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:  $13, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(299=13\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{299}(1,·)$, $\chi_{299}(194,·)$, $\chi_{299}(131,·)$, $\chi_{299}(196,·)$, $\chi_{299}(261,·)$, $\chi_{299}(129,·)$, $\chi_{299}(144,·)$, $\chi_{299}(209,·)$, $\chi_{299}(90,·)$, $\chi_{299}(27,·)$, $\chi_{299}(272,·)$, $\chi_{299}(155,·)$, $\chi_{299}(38,·)$, $\chi_{299}(103,·)$, $\chi_{299}(168,·)$, $\chi_{299}(105,·)$, $\chi_{299}(298,·)$, $\chi_{299}(51,·)$, $\chi_{299}(181,·)$, $\chi_{299}(118,·)$, $\chi_{299}(248,·)$, $\chi_{299}(170,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{59541067} a^{12} - \frac{11473253}{59541067} a^{11} + \frac{36}{59541067} a^{10} - \frac{21370947}{59541067} a^{9} + \frac{486}{59541067} a^{8} - \frac{18287096}{59541067} a^{7} + \frac{3024}{59541067} a^{6} + \frac{23074880}{59541067} a^{5} + \frac{8505}{59541067} a^{4} - \frac{27106629}{59541067} a^{3} + \frac{8748}{59541067} a^{2} - \frac{4355764}{59541067} a + \frac{1458}{59541067}$, $\frac{1}{59541067} a^{13} + \frac{39}{59541067} a^{11} - \frac{25121308}{59541067} a^{10} + \frac{585}{59541067} a^{9} + \frac{20394631}{59541067} a^{8} + \frac{4212}{59541067} a^{7} + \frac{5749891}{59541067} a^{6} + \frac{14742}{59541067} a^{5} + \frac{24642390}{59541067} a^{4} + \frac{22113}{59541067} a^{3} - \frac{22577482}{59541067} a^{2} + \frac{9477}{59541067} a - \frac{3036953}{59541067}$, $\frac{1}{59541067} a^{14} + \frac{5548090}{59541067} a^{11} - \frac{819}{59541067} a^{10} + \frac{20286626}{59541067} a^{9} - \frac{14742}{59541067} a^{8} + \frac{4453831}{59541067} a^{7} - \frac{103194}{59541067} a^{6} + \frac{17838075}{59541067} a^{5} - \frac{309582}{59541067} a^{4} + \frac{22382910}{59541067} a^{3} - \frac{331695}{59541067} a^{2} - \frac{11785358}{59541067} a - \frac{56862}{59541067}$, $\frac{1}{59541067} a^{15} - \frac{945}{59541067} a^{11} - \frac{821413}{59541067} a^{10} - \frac{18900}{59541067} a^{9} - \frac{12569894}{59541067} a^{8} - \frac{153090}{59541067} a^{7} - \frac{28546258}{59541067} a^{6} - \frac{571536}{59541067} a^{5} - \frac{7597476}{59541067} a^{4} - \frac{893025}{59541067} a^{3} - \frac{20507073}{59541067} a^{2} - \frac{393660}{59541067} a + \frac{8469892}{59541067}$, $\frac{1}{59541067} a^{16} - \frac{6571304}{59541067} a^{11} + \frac{15120}{59541067} a^{10} - \frac{23693096}{59541067} a^{9} + \frac{306180}{59541067} a^{8} + \frac{16598519}{59541067} a^{7} + \frac{2286144}{59541067} a^{6} + \frac{6133602}{59541067} a^{5} + \frac{7144200}{59541067} a^{4} + \frac{25928399}{59541067} a^{3} + \frac{7873200}{59541067} a^{2} + \frac{606535}{59541067} a + \frac{1377810}{59541067}$, $\frac{1}{59541067} a^{17} + \frac{18360}{59541067} a^{11} - \frac{25290420}{59541067} a^{10} + \frac{413100}{59541067} a^{9} - \frac{4965355}{59541067} a^{8} + \frac{3569184}{59541067} a^{7} - \frac{8959480}{59541067} a^{6} + \frac{13880160}{59541067} a^{5} + \frac{5807006}{59541067} a^{4} + \frac{22307400}{59541067} a^{3} + \frac{29244272}{59541067} a^{2} + \frac{10038330}{59541067} a - \frac{5150555}{59541067}$, $\frac{1}{59541067} a^{18} + \frac{26880681}{59541067} a^{11} - \frac{247860}{59541067} a^{10} - \frac{10009965}{59541067} a^{9} - \frac{5353776}{59541067} a^{8} - \frac{9953733}{59541067} a^{7} + \frac{17900587}{59541067} a^{6} - \frac{14298089}{59541067} a^{5} - \frac{14762266}{59541067} a^{4} + \frac{3173659}{59541067} a^{3} + \frac{28048251}{59541067} a^{2} + \frac{3023504}{59541067} a - \frac{26768880}{59541067}$, $\frac{1}{59541067} a^{19} - \frac{313956}{59541067} a^{11} - \frac{25057409}{59541067} a^{10} - \frac{7534944}{59541067} a^{9} + \frac{25070041}{59541067} a^{8} - \frac{8273429}{59541067} a^{7} - \frac{27920978}{59541067} a^{6} + \frac{26447351}{59541067} a^{5} + \frac{20679034}{59541067} a^{4} - \frac{28245161}{59541067} a^{3} - \frac{21500301}{59541067} a^{2} - \frac{24820287}{59541067} a - \frac{14010812}{59541067}$, $\frac{1}{59541067} a^{20} - \frac{6204911}{59541067} a^{11} + \frac{3767472}{59541067} a^{10} - \frac{7749262}{59541067} a^{9} + \frac{25227053}{59541067} a^{8} - \frac{4965145}{59541067} a^{7} + \frac{23193223}{59541067} a^{6} - \frac{22540777}{59541067} a^{5} + \frac{22143671}{59541067} a^{4} + \frac{13473819}{59541067} a^{3} - \frac{17222281}{59541067} a^{2} + \frac{6973660}{59541067} a - \frac{18580688}{59541067}$, $\frac{1}{59541067} a^{21} + \frac{4944807}{59541067} a^{11} - \frac{22536734}{59541067} a^{10} + \frac{4538041}{59541067} a^{9} - \frac{25972816}{59541067} a^{8} + \frac{13089347}{59541067} a^{7} - \frac{14326018}{59541067} a^{6} + \frac{28639389}{59541067} a^{5} - \frac{26684555}{59541067} a^{4} - \frac{11808752}{59541067} a^{3} - \frac{13918016}{59541067} a^{2} - \frac{27240784}{59541067} a - \frac{3481946}{59541067}$

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

Class group and class number

$C_{138344}$, which has order $138344$ (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:  \( 1038656.82438 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{22}$ (as 22T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-299}) \), \(\Q(\zeta_{23})^+\)

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 $22$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ R $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$ $22$ $22$

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
13Data not computed
23Data not computed