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Properties

Label 20.0.11244621349...5625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 11^{16}\cdot 1583^{2}$
Root discriminant $31.81$
Ramified primes $5, 11, 1583$
Class number $33$ (GRH)
Class group $[33]$ (GRH)
Galois group $C_2^2\times C_2^4:C_5$ (as 20T86)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, 2048, 5120, 7680, 10880, 11872, 11440, 8624, 5596, 2068, 1299, -1034, 1399, -1078, 715, -371, 170, -60, 20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 20*x^18 - 60*x^17 + 170*x^16 - 371*x^15 + 715*x^14 - 1078*x^13 + 1399*x^12 - 1034*x^11 + 1299*x^10 + 2068*x^9 + 5596*x^8 + 8624*x^7 + 11440*x^6 + 11872*x^5 + 10880*x^4 + 7680*x^3 + 5120*x^2 + 2048*x + 1024)
 
gp: K = bnfinit(x^20 - 4*x^19 + 20*x^18 - 60*x^17 + 170*x^16 - 371*x^15 + 715*x^14 - 1078*x^13 + 1399*x^12 - 1034*x^11 + 1299*x^10 + 2068*x^9 + 5596*x^8 + 8624*x^7 + 11440*x^6 + 11872*x^5 + 10880*x^4 + 7680*x^3 + 5120*x^2 + 2048*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 20 x^{18} - 60 x^{17} + 170 x^{16} - 371 x^{15} + 715 x^{14} - 1078 x^{13} + 1399 x^{12} - 1034 x^{11} + 1299 x^{10} + 2068 x^{9} + 5596 x^{8} + 8624 x^{7} + 11440 x^{6} + 11872 x^{5} + 10880 x^{4} + 7680 x^{3} + 5120 x^{2} + 2048 x + 1024 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1124462134942353310118447265625=5^{10}\cdot 11^{16}\cdot 1583^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.81$
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:  $5, 11, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} - \frac{3}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{4} a^{6} + \frac{3}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{14} - \frac{1}{4} a^{11} - \frac{3}{8} a^{10} + \frac{5}{16} a^{9} + \frac{7}{16} a^{8} + \frac{1}{8} a^{7} - \frac{5}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a$, $\frac{1}{352} a^{15} - \frac{1}{44} a^{14} - \frac{1}{22} a^{13} + \frac{9}{88} a^{12} + \frac{21}{176} a^{11} + \frac{117}{352} a^{10} + \frac{13}{32} a^{9} - \frac{67}{176} a^{8} + \frac{115}{352} a^{7} - \frac{5}{16} a^{6} - \frac{81}{352} a^{5} + \frac{3}{22} a^{4} + \frac{9}{22} a^{3} - \frac{3}{22} a^{2} - \frac{4}{11} a - \frac{1}{11}$, $\frac{1}{704} a^{16} + \frac{1}{88} a^{14} - \frac{1}{176} a^{13} - \frac{1}{32} a^{12} + \frac{101}{704} a^{11} + \frac{199}{704} a^{10} + \frac{109}{352} a^{9} + \frac{9}{64} a^{8} - \frac{79}{352} a^{7} + \frac{183}{704} a^{6} + \frac{3}{11} a^{5} + \frac{3}{8} a^{4} - \frac{5}{88} a^{3} + \frac{3}{11} a^{2} - \frac{4}{11}$, $\frac{1}{1408} a^{17} - \frac{7}{352} a^{14} - \frac{35}{704} a^{13} + \frac{15}{128} a^{12} + \frac{215}{1408} a^{11} + \frac{257}{704} a^{10} - \frac{39}{128} a^{9} + \frac{61}{704} a^{8} + \frac{29}{128} a^{7} + \frac{13}{176} a^{6} + \frac{13}{88} a^{5} - \frac{21}{88} a^{4} - \frac{5}{88} a^{3} - \frac{21}{44} a^{2} + \frac{1}{22} a + \frac{2}{11}$, $\frac{1}{6387037834496} a^{18} - \frac{678809227}{3193518917248} a^{17} - \frac{13070053}{99797466164} a^{16} + \frac{970007751}{1596759458624} a^{15} - \frac{8544918637}{290319901568} a^{14} + \frac{265963632425}{6387037834496} a^{13} - \frac{62667930565}{580639803136} a^{12} + \frac{42248074367}{798379729312} a^{11} + \frac{646907335551}{6387037834496} a^{10} - \frac{3252967317}{199594932328} a^{9} + \frac{1334941312627}{6387037834496} a^{8} + \frac{977629712595}{3193518917248} a^{7} - \frac{139570905211}{399189864656} a^{6} - \frac{32676868903}{72579975392} a^{5} + \frac{137823375773}{399189864656} a^{4} - \frac{11996838613}{99797466164} a^{3} + \frac{21249601471}{49898733082} a^{2} - \frac{8161991147}{24949366541} a + \frac{6804372691}{24949366541}$, $\frac{1}{855863069822464} a^{19} - \frac{3}{106982883727808} a^{18} + \frac{35139645163}{213965767455616} a^{17} - \frac{84953237291}{213965767455616} a^{16} + \frac{123668151285}{427931534911232} a^{15} - \frac{6339237311419}{855863069822464} a^{14} + \frac{30386071198935}{855863069822464} a^{13} - \frac{26479114475925}{427931534911232} a^{12} + \frac{199336009536743}{855863069822464} a^{11} - \frac{182409703994411}{427931534911232} a^{10} + \frac{29114122733481}{77805733620224} a^{9} + \frac{9524104860697}{106982883727808} a^{8} + \frac{8845165422071}{19451433405056} a^{7} - \frac{313285936587}{1162857431824} a^{6} + \frac{6131531930459}{26745720931952} a^{5} - \frac{3849451445879}{13372860465976} a^{4} - \frac{5045058786959}{13372860465976} a^{3} - \frac{250437950373}{1671607558247} a^{2} - \frac{232845434073}{3343215116494} a - \frac{538903061177}{1671607558247}$

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

Class group and class number

$C_{33}$, which has order $33$ (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:  $9$
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:  \( 140644.599182 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_2^4:C_5$ (as 20T86):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 32 conjugacy class representatives for $C_2^2\times C_2^4:C_5$
Character table for $C_2^2\times C_2^4:C_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.339330108623.1, 10.0.1060406589446875.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
1583Data not computed