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Properties

Label 20.0.12719384508...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 13^{10}$
Root discriminant $63.86$
Ramified primes $2, 3, 5, 13$
Class number Not computed
Class group Not computed
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![3540446599, -86405200, 4442944605, -46929210, 2529545715, -2398998, 828979920, 1618410, 171185925, 273710, 23259353, 14710, 2110675, 170, 126620, -4, 4815, 0, 105, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 105*x^18 + 4815*x^16 - 4*x^15 + 126620*x^14 + 170*x^13 + 2110675*x^12 + 14710*x^11 + 23259353*x^10 + 273710*x^9 + 171185925*x^8 + 1618410*x^7 + 828979920*x^6 - 2398998*x^5 + 2529545715*x^4 - 46929210*x^3 + 4442944605*x^2 - 86405200*x + 3540446599)
 
gp: K = bnfinit(x^20 + 105*x^18 + 4815*x^16 - 4*x^15 + 126620*x^14 + 170*x^13 + 2110675*x^12 + 14710*x^11 + 23259353*x^10 + 273710*x^9 + 171185925*x^8 + 1618410*x^7 + 828979920*x^6 - 2398998*x^5 + 2529545715*x^4 - 46929210*x^3 + 4442944605*x^2 - 86405200*x + 3540446599, 1)
 

Normalized defining polynomial

\( x^{20} + 105 x^{18} + 4815 x^{16} - 4 x^{15} + 126620 x^{14} + 170 x^{13} + 2110675 x^{12} + 14710 x^{11} + 23259353 x^{10} + 273710 x^{9} + 171185925 x^{8} + 1618410 x^{7} + 828979920 x^{6} - 2398998 x^{5} + 2529545715 x^{4} - 46929210 x^{3} + 4442944605 x^{2} - 86405200 x + 3540446599 \)

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:  \(1271938450811187656250000000000000000=2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.86$
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:  $2, 3, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} - \frac{3}{14} a^{16} + \frac{3}{14} a^{15} - \frac{1}{7} a^{14} + \frac{1}{14} a^{13} - \frac{3}{14} a^{12} - \frac{1}{14} a^{11} - \frac{3}{14} a^{10} + \frac{3}{14} a^{9} - \frac{3}{14} a^{8} - \frac{3}{14} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{5} - \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{14} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{734235824885546826067692700464652279750367527357060875296008473748114} a^{19} + \frac{10077452403613906497503124441741096431939070346751007834582254916710}{367117912442773413033846350232326139875183763678530437648004236874057} a^{18} - \frac{538999350711238538709193964421779743759251274359077766758127885059}{5520570111921404707275884965899641201130582912459104325533898298858} a^{17} - \frac{942399458856344329081055810415242499602327874793008999621900278363}{38643990783449832950931194761297488407914080387213730278737288092006} a^{16} + \frac{51952639256696544776015276357868891718491476739191843537643435196748}{367117912442773413033846350232326139875183763678530437648004236874057} a^{15} - \frac{41933087033249016827050759816207594687868553088064845122720141120571}{367117912442773413033846350232326139875183763678530437648004236874057} a^{14} - \frac{144618500788005448310000576645382527204785270693411415215240159142631}{734235824885546826067692700464652279750367527357060875296008473748114} a^{13} + \frac{113966467956848379451929316274048500795702250412741783527058023396959}{734235824885546826067692700464652279750367527357060875296008473748114} a^{12} - \frac{3190540047550293031808379367519118240856783091005388881652035057002}{19321995391724916475465597380648744203957040193606865139368644046003} a^{11} + \frac{83068269938566569217236671831939629547533658552950972649682412048061}{734235824885546826067692700464652279750367527357060875296008473748114} a^{10} - \frac{26398116037714025322946857318187429622292896456763760685064416416671}{734235824885546826067692700464652279750367527357060875296008473748114} a^{9} + \frac{21288751028112588742974604373091115533783645000640248649444210318341}{104890832126506689438241814352093182821481075336722982185144067678302} a^{8} + \frac{15692175501618277245958923188626755126298039064635457162836167661283}{104890832126506689438241814352093182821481075336722982185144067678302} a^{7} + \frac{36351106377265069468711066931018249949774140602015188041072917187239}{734235824885546826067692700464652279750367527357060875296008473748114} a^{6} - \frac{94060170907791207199080205835353352244079450790931739893117948961862}{367117912442773413033846350232326139875183763678530437648004236874057} a^{5} + \frac{337672878577890778016774979063514211999970728779325447069835862005}{2760285055960702353637942482949820600565291456229552162766949149429} a^{4} - \frac{7598296588643740831987663420669984648184653948408742407531549935381}{19321995391724916475465597380648744203957040193606865139368644046003} a^{3} - \frac{234956047971729980519736806666456103820495259504552332442817203601719}{734235824885546826067692700464652279750367527357060875296008473748114} a^{2} - \frac{5899069682011644292019080212239555855265140990944478805984049648166}{19321995391724916475465597380648744203957040193606865139368644046003} a + \frac{6731101851128430898900843188867582035280892461737521491339951813960}{367117912442773413033846350232326139875183763678530437648004236874057}$

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

\(\Q(\sqrt{-195}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-39})\), 5.1.50000.1, 10.0.1127802487500000000.1, 10.0.225560497500000000.1, 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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$