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Properties

Label 20.0.40527225939...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{22}\cdot 11^{10}$
Root discriminant $33.91$
Ramified primes $2, 5, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3980, 2100, -8980, -19480, 3000, -7630, 63020, 13030, 82300, 8340, 51049, 1410, 18305, 30, 3980, -4, 510, 0, 35, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 35*x^18 + 510*x^16 - 4*x^15 + 3980*x^14 + 30*x^13 + 18305*x^12 + 1410*x^11 + 51049*x^10 + 8340*x^9 + 82300*x^8 + 13030*x^7 + 63020*x^6 - 7630*x^5 + 3000*x^4 - 19480*x^3 - 8980*x^2 + 2100*x + 3980)
 
gp: K = bnfinit(x^20 + 35*x^18 + 510*x^16 - 4*x^15 + 3980*x^14 + 30*x^13 + 18305*x^12 + 1410*x^11 + 51049*x^10 + 8340*x^9 + 82300*x^8 + 13030*x^7 + 63020*x^6 - 7630*x^5 + 3000*x^4 - 19480*x^3 - 8980*x^2 + 2100*x + 3980, 1)
 

Normalized defining polynomial

\( x^{20} + 35 x^{18} + 510 x^{16} - 4 x^{15} + 3980 x^{14} + 30 x^{13} + 18305 x^{12} + 1410 x^{11} + 51049 x^{10} + 8340 x^{9} + 82300 x^{8} + 13030 x^{7} + 63020 x^{6} - 7630 x^{5} + 3000 x^{4} - 19480 x^{3} - 8980 x^{2} + 2100 x + 3980 \)

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:  \(4052722593906250000000000000000=2^{16}\cdot 5^{22}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.91$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} - \frac{3}{14} a^{16} + \frac{3}{14} a^{15} - \frac{1}{7} a^{14} + \frac{1}{14} a^{13} - \frac{3}{14} a^{12} - \frac{1}{14} a^{11} - \frac{3}{14} a^{10} + \frac{3}{14} a^{9} + \frac{2}{7} a^{8} + \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{14} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{320501857394001717824096380066806276523786} a^{19} - \frac{3472974015188267035957169348630392633173}{320501857394001717824096380066806276523786} a^{18} - \frac{23473831755324190274142377154159094652353}{160250928697000858912048190033403138261893} a^{17} - \frac{11611377129659000863220424856365170751261}{160250928697000858912048190033403138261893} a^{16} - \frac{60127057225915776090269203884911370859735}{320501857394001717824096380066806276523786} a^{15} + \frac{4474345534081613933917037813894991546477}{22892989813857265558864027147629019751699} a^{14} + \frac{2033593782000757785543067608818761422485}{11051788186000059235313668278165733673234} a^{13} - \frac{8388324744773724364679810866340103972706}{160250928697000858912048190033403138261893} a^{12} - \frac{1144415476971356979852960446331093773755}{22892989813857265558864027147629019751699} a^{11} + \frac{748993510074993969634301785606475581750}{160250928697000858912048190033403138261893} a^{10} + \frac{59623653934828985256659540080810248388433}{160250928697000858912048190033403138261893} a^{9} - \frac{13019821856267788885486968579854585109223}{160250928697000858912048190033403138261893} a^{8} + \frac{58061520566417020646507830494121720996278}{160250928697000858912048190033403138261893} a^{7} - \frac{13939306114925642950045684192781885971391}{320501857394001717824096380066806276523786} a^{6} + \frac{143515431927225613574868175647743288721637}{320501857394001717824096380066806276523786} a^{5} + \frac{29958542575895706377794075861930431541370}{160250928697000858912048190033403138261893} a^{4} - \frac{20240765965359977041293136142240534918109}{160250928697000858912048190033403138261893} a^{3} + \frac{46060618940438376678000705184260822968145}{160250928697000858912048190033403138261893} a^{2} - \frac{6189531493984111896385066585419662839831}{22892989813857265558864027147629019751699} a + \frac{44302669339470803175335144613622359329867}{160250928697000858912048190033403138261893}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 10796039.931608813 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{-55}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-11})\), 5.1.50000.1, 10.0.2013137500000000.1, 10.0.402627500000000.2, 10.2.12500000000.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$11$11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$