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Properties

Label 20.4.16599951744...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{28}\cdot 5^{22}\cdot 11^{10}$
Root discriminant $51.40$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (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![38901449, 3993000, -25437500, 29375000, 3025, 5616, 2253500, 440, -10990, 70000, -140648, -400, 2390, -8280, 0, -8, -175, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 175*x^16 - 8*x^15 - 8280*x^13 + 2390*x^12 - 400*x^11 - 140648*x^10 + 70000*x^9 - 10990*x^8 + 440*x^7 + 2253500*x^6 + 5616*x^5 + 3025*x^4 + 29375000*x^3 - 25437500*x^2 + 3993000*x + 38901449)
 
gp: K = bnfinit(x^20 - 175*x^16 - 8*x^15 - 8280*x^13 + 2390*x^12 - 400*x^11 - 140648*x^10 + 70000*x^9 - 10990*x^8 + 440*x^7 + 2253500*x^6 + 5616*x^5 + 3025*x^4 + 29375000*x^3 - 25437500*x^2 + 3993000*x + 38901449, 1)
 

Normalized defining polynomial

\( x^{20} - 175 x^{16} - 8 x^{15} - 8280 x^{13} + 2390 x^{12} - 400 x^{11} - 140648 x^{10} + 70000 x^{9} - 10990 x^{8} + 440 x^{7} + 2253500 x^{6} + 5616 x^{5} + 3025 x^{4} + 29375000 x^{3} - 25437500 x^{2} + 3993000 x + 38901449 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(16599951744640000000000000000000000=2^{28}\cdot 5^{22}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.40$
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:  $2, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{1812} a^{17} - \frac{73}{1812} a^{16} - \frac{55}{604} a^{15} - \frac{15}{604} a^{14} - \frac{65}{604} a^{13} + \frac{451}{1812} a^{12} - \frac{55}{1812} a^{11} - \frac{41}{604} a^{10} - \frac{811}{1812} a^{9} + \frac{679}{1812} a^{8} - \frac{779}{1812} a^{7} + \frac{267}{604} a^{6} + \frac{63}{604} a^{5} + \frac{667}{1812} a^{4} - \frac{617}{1812} a^{3} - \frac{113}{1812} a^{2} + \frac{37}{453} a - \frac{13}{453}$, $\frac{1}{1812} a^{18} - \frac{29}{906} a^{16} + \frac{47}{604} a^{15} + \frac{12}{151} a^{14} + \frac{259}{1812} a^{13} + \frac{21}{151} a^{12} - \frac{61}{1812} a^{11} + \frac{44}{453} a^{10} - \frac{29}{604} a^{9} - \frac{34}{453} a^{8} - \frac{347}{1812} a^{7} + \frac{113}{302} a^{6} + \frac{421}{1812} a^{5} - \frac{425}{906} a^{4} + \frac{599}{1812} a^{3} + \frac{53}{1812} a^{2} - \frac{191}{604} a - \frac{43}{453}$, $\frac{1}{60585827856464805668903647869051526727612657475684451896890453577551964} a^{19} + \frac{1555782319726413593067778350865089443900162742487454414283362927699}{10097637976077467611483941311508587787935442912614075316148408929591994} a^{18} - \frac{4100776436416737502403656298782318978979799184723249515432933877361}{30292913928232402834451823934525763363806328737842225948445226788775982} a^{17} + \frac{48169039275766375081775222913908862541191229346462834041026396271339}{5048818988038733805741970655754293893967721456307037658074204464795997} a^{16} - \frac{995136572557658595166781330447636262967211572820418025156985919313997}{10097637976077467611483941311508587787935442912614075316148408929591994} a^{15} + \frac{993051738499713850141008938018600420095957033073239872565369935388078}{15146456964116201417225911967262881681903164368921112974222613394387991} a^{14} + \frac{820199625619014082693264394673229695305193826859732428615955340166739}{5048818988038733805741970655754293893967721456307037658074204464795997} a^{13} + \frac{6585404314727143727972496175380123038354982484626418404100042974328141}{30292913928232402834451823934525763363806328737842225948445226788775982} a^{12} - \frac{1032079429338616321180091379731918781946037056042724762425651307997934}{15146456964116201417225911967262881681903164368921112974222613394387991} a^{11} - \frac{405280477563574837596767160027813041640675831740834761097137780178963}{10097637976077467611483941311508587787935442912614075316148408929591994} a^{10} - \frac{3969233840350370625498095191503873859571056091071054008224520237954988}{15146456964116201417225911967262881681903164368921112974222613394387991} a^{9} - \frac{2716045749741305523282125996385897762114571500632346483957561216673067}{15146456964116201417225911967262881681903164368921112974222613394387991} a^{8} - \frac{339441471664595701902345194920138671895785278651100201153050650358927}{5048818988038733805741970655754293893967721456307037658074204464795997} a^{7} + \frac{2244067008086807740539744943661633660361256010935451045890548729046684}{15146456964116201417225911967262881681903164368921112974222613394387991} a^{6} - \frac{3760018846254293862756408027737004948098238393303982533389332035265949}{30292913928232402834451823934525763363806328737842225948445226788775982} a^{5} - \frac{6070738467869437226411291657358639264275305170040248598147083406348775}{30292913928232402834451823934525763363806328737842225948445226788775982} a^{4} + \frac{20218444646669913914425062171545112932580703381701830463519174136722185}{60585827856464805668903647869051526727612657475684451896890453577551964} a^{3} - \frac{416810590131825625730529006740655156951789202705660503565038728062110}{5048818988038733805741970655754293893967721456307037658074204464795997} a^{2} + \frac{3794501527070394339687188027429499167540454420722575169403106048753967}{15146456964116201417225911967262881681903164368921112974222613394387991} a - \frac{145770325700060392994843970635655419950982493703944568946073785719007}{5048818988038733805741970655754293893967721456307037658074204464795997}$

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

Class group and class number

Trivial group, which has 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:  $11$
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:  \( 3363644183.928467 \) (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, 10.2.25768160000000000.1, 10.2.128840800000000000.1, 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 ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
$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$11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11.10.5.1$x^{10} - 242 x^{6} - 1331 x^{4} - 131769 x^{2} - 4026275$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$