↑

Properties

Label 20.0.79861992703...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{22}\cdot 59^{10}$
Root discriminant $78.54$
Ramified primes $2, 5, 59$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T13)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![290388198524, -1998540300, 227754848780, -577697560, 79491889440, -14003998, 16102101620, 8435110, 2088869500, 907860, 181224553, 32610, 10654025, 270, 419420, -4, 10590, 0, 155, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 155*x^18 + 10590*x^16 - 4*x^15 + 419420*x^14 + 270*x^13 + 10654025*x^12 + 32610*x^11 + 181224553*x^10 + 907860*x^9 + 2088869500*x^8 + 8435110*x^7 + 16102101620*x^6 - 14003998*x^5 + 79491889440*x^4 - 577697560*x^3 + 227754848780*x^2 - 1998540300*x + 290388198524)
 
gp: K = bnfinit(x^20 + 155*x^18 + 10590*x^16 - 4*x^15 + 419420*x^14 + 270*x^13 + 10654025*x^12 + 32610*x^11 + 181224553*x^10 + 907860*x^9 + 2088869500*x^8 + 8435110*x^7 + 16102101620*x^6 - 14003998*x^5 + 79491889440*x^4 - 577697560*x^3 + 227754848780*x^2 - 1998540300*x + 290388198524, 1)
 

Normalized defining polynomial

\( x^{20} + 155 x^{18} + 10590 x^{16} - 4 x^{15} + 419420 x^{14} + 270 x^{13} + 10654025 x^{12} + 32610 x^{11} + 181224553 x^{10} + 907860 x^{9} + 2088869500 x^{8} + 8435110 x^{7} + 16102101620 x^{6} - 14003998 x^{5} + 79491889440 x^{4} - 577697560 x^{3} + 227754848780 x^{2} - 1998540300 x + 290388198524 \)

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:  \(79861992703225218906250000000000000000=2^{16}\cdot 5^{22}\cdot 59^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.54$
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:  $2, 5, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} + \frac{1}{7} a^{15} - \frac{3}{14} a^{13} + \frac{3}{14} a^{12} - \frac{3}{14} a^{11} - \frac{3}{14} a^{10} + \frac{1}{14} a^{9} + \frac{5}{14} a^{8} + \frac{5}{14} a^{7} + \frac{1}{7} a^{6} + \frac{5}{14} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{181074370959624157641003171514303164227655536915444759448307025112499517817514} a^{19} - \frac{5113927251925331245408920803407476940295330353476009167792916927842562208295}{181074370959624157641003171514303164227655536915444759448307025112499517817514} a^{18} + \frac{18095641116955830905817961672296526986904004264863726741575010822770498841422}{90537185479812078820501585757151582113827768457722379724153512556249758908757} a^{17} - \frac{37451076670906903440790762788693219852835362044289353625732121887708179365713}{181074370959624157641003171514303164227655536915444759448307025112499517817514} a^{16} + \frac{4081780023634790315135483137095700681394789068153973955072311930493518558872}{90537185479812078820501585757151582113827768457722379724153512556249758908757} a^{15} + \frac{4938462151612733992698151283492417209957084531473931949195488548995515822552}{90537185479812078820501585757151582113827768457722379724153512556249758908757} a^{14} - \frac{441346886343181247437087544854989821495884637822572178094086033906802668995}{181074370959624157641003171514303164227655536915444759448307025112499517817514} a^{13} - \frac{43907902071603764454560103787165311502763891488839779393216126017433033201565}{181074370959624157641003171514303164227655536915444759448307025112499517817514} a^{12} - \frac{6825711084670875264748299990082136799012738472285986682841170977500236739066}{90537185479812078820501585757151582113827768457722379724153512556249758908757} a^{11} + \frac{33130618406313334891041475122450529939925604463874734023430938262839842628393}{181074370959624157641003171514303164227655536915444759448307025112499517817514} a^{10} + \frac{1623260690558116128439874460320945887697894235940568782672876392988611452725}{12933883639973154117214512251021654587689681208246054246307644650892822701251} a^{9} + \frac{17497662320787504698894128731643627252988550661089564526479310408142642948457}{90537185479812078820501585757151582113827768457722379724153512556249758908757} a^{8} - \frac{14033662058887721508414900571762249913874184414345344481294827783051362552389}{181074370959624157641003171514303164227655536915444759448307025112499517817514} a^{7} + \frac{42163587673036686247721238580211840321895886714795654643533750063261983420579}{90537185479812078820501585757151582113827768457722379724153512556249758908757} a^{6} + \frac{15526039923242539637415853821158812247128686429202409597861770872317271434521}{181074370959624157641003171514303164227655536915444759448307025112499517817514} a^{5} - \frac{9661517292059251355183331046044041233044680115032347972525979111824552900316}{90537185479812078820501585757151582113827768457722379724153512556249758908757} a^{4} + \frac{6322377475860559752613387240423171806830935314861759516608508729845866343137}{12933883639973154117214512251021654587689681208246054246307644650892822701251} a^{3} + \frac{12362071571478360842149135409844776384887241388045130653978299130236132589417}{90537185479812078820501585757151582113827768457722379724153512556249758908757} a^{2} - \frac{44516681511418578796615310373262397822135668154503925805294230268947535797674}{90537185479812078820501585757151582113827768457722379724153512556249758908757} a + \frac{936383120198734072553179490869650399308108052885753107581789468096170030175}{90537185479812078820501585757151582113827768457722379724153512556249758908757}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

\(\Q(\sqrt{-295}) \), \(\Q(\sqrt{-59}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-59})\), 5.1.50000.1, 10.0.8936553737500000000.1, 10.0.1787310747500000000.1, 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ R

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}$
$59$59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.2.1.2$x^{2} + 177$$2$$1$$1$$C_2$$[\ ]_{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
59.4.2.1$x^{4} + 177 x^{2} + 13924$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$