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Properties

Label 22.0.42024982875...6803.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{11}\cdot 421^{2}\cdot 115692385433^{2}$
Root discriminant $30.40$
Ramified primes $3, 421, 115692385433$
Class number $28$ (GRH)
Class group $[28]$ (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, -13, 152, -323, 1047, -1619, 4182, -5447, 8780, -7261, 8435, -5457, 5421, -2774, 2270, -925, 675, -221, 133, -32, 17, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 + 17*x^20 - 32*x^19 + 133*x^18 - 221*x^17 + 675*x^16 - 925*x^15 + 2270*x^14 - 2774*x^13 + 5421*x^12 - 5457*x^11 + 8435*x^10 - 7261*x^9 + 8780*x^8 - 5447*x^7 + 4182*x^6 - 1619*x^5 + 1047*x^4 - 323*x^3 + 152*x^2 - 13*x + 1)
 
gp: K = bnfinit(x^22 - 3*x^21 + 17*x^20 - 32*x^19 + 133*x^18 - 221*x^17 + 675*x^16 - 925*x^15 + 2270*x^14 - 2774*x^13 + 5421*x^12 - 5457*x^11 + 8435*x^10 - 7261*x^9 + 8780*x^8 - 5447*x^7 + 4182*x^6 - 1619*x^5 + 1047*x^4 - 323*x^3 + 152*x^2 - 13*x + 1, 1)
 

Normalized defining polynomial

\( x^{22} - 3 x^{21} + 17 x^{20} - 32 x^{19} + 133 x^{18} - 221 x^{17} + 675 x^{16} - 925 x^{15} + 2270 x^{14} - 2774 x^{13} + 5421 x^{12} - 5457 x^{11} + 8435 x^{10} - 7261 x^{9} + 8780 x^{8} - 5447 x^{7} + 4182 x^{6} - 1619 x^{5} + 1047 x^{4} - 323 x^{3} + 152 x^{2} - 13 x + 1 \)

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:  \(-420249828754162766851850612806803=-\,3^{11}\cdot 421^{2}\cdot 115692385433^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.40$
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:  $3, 421, 115692385433$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{4253743685312477080655337611} a^{21} - \frac{1712911819778504924691978793}{4253743685312477080655337611} a^{20} - \frac{294140067584464833186878594}{4253743685312477080655337611} a^{19} + \frac{503197510327270849306729834}{4253743685312477080655337611} a^{18} - \frac{932160094380618336457864598}{4253743685312477080655337611} a^{17} - \frac{627034878672009367176370103}{4253743685312477080655337611} a^{16} + \frac{1237697758937029198321166849}{4253743685312477080655337611} a^{15} + \frac{530301279173583846600880167}{4253743685312477080655337611} a^{14} + \frac{454026986587666290688716583}{4253743685312477080655337611} a^{13} + \frac{185569043304869617570824490}{4253743685312477080655337611} a^{12} + \frac{1610174167088723084043038044}{4253743685312477080655337611} a^{11} + \frac{1711436160126445492346618121}{4253743685312477080655337611} a^{10} - \frac{733338684787654531446587505}{4253743685312477080655337611} a^{9} - \frac{1969550223964527169481586866}{4253743685312477080655337611} a^{8} - \frac{526889978078387271700166312}{4253743685312477080655337611} a^{7} - \frac{786368300170961422309468271}{4253743685312477080655337611} a^{6} - \frac{1153892755552653877642611115}{4253743685312477080655337611} a^{5} - \frac{1116722343893067679893362848}{4253743685312477080655337611} a^{4} - \frac{1507717671840545761113601009}{4253743685312477080655337611} a^{3} + \frac{1481550378589452292650829537}{4253743685312477080655337611} a^{2} - \frac{1708449677453076130016473897}{4253743685312477080655337611} a - \frac{382140220972787477870941127}{4253743685312477080655337611}$

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

Class group and class number

$C_{28}$, which has order $28$ (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:  \( -\frac{355574385981467340978346053}{4253743685312477080655337611} a^{21} + \frac{1066603240841898135085181019}{4253743685312477080655337611} a^{20} - \frac{6057952249385138720228760690}{4253743685312477080655337611} a^{19} + \frac{11412711042433525673913199261}{4253743685312477080655337611} a^{18} - \frac{47502142730751997704433579856}{4253743685312477080655337611} a^{17} + \frac{78916805441351165819686067916}{4253743685312477080655337611} a^{16} - \frac{241599460599860975030386918529}{4253743685312477080655337611} a^{15} + \frac{331141186718285117979891484590}{4253743685312477080655337611} a^{14} - \frac{814844410712891935559820916607}{4253743685312477080655337611} a^{13} + \frac{994986911456564060055123706148}{4253743685312477080655337611} a^{12} - \frac{1952018959275358562140386125359}{4253743685312477080655337611} a^{11} + \frac{1964482766419870236321552181681}{4253743685312477080655337611} a^{10} - \frac{3053621162181021376123608868210}{4253743685312477080655337611} a^{9} + \frac{2622498864677571826497306599694}{4253743685312477080655337611} a^{8} - \frac{3196738985237440790596874012996}{4253743685312477080655337611} a^{7} + \frac{1980818918402080038077995029655}{4253743685312477080655337611} a^{6} - \frac{1552464917184013268553570305375}{4253743685312477080655337611} a^{5} + \frac{590382949711489814684400051988}{4253743685312477080655337611} a^{4} - \frac{384947509565447967790860565418}{4253743685312477080655337611} a^{3} + \frac{103525016849980206157292878556}{4253743685312477080655337611} a^{2} - \frac{56318499580264956183856614738}{4253743685312477080655337611} a + \frac{4818276375486568627174599919}{4253743685312477080655337611} \) (order $6$)
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:  \( 1227458.81752 \) (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
The 112 conjugacy class representatives for t22n47 are not computed
Character table for t22n47 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.11.48706494267293.1

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
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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ $22$ $18{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
421Data not computed
115692385433Data not computed