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Properties

Label 20.0.90346941218...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 5^{22}\cdot 13^{10}$
Root discriminant $79.03$
Ramified primes $2, 5, 13$
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![551525305475, -564365350, 1570569425, -2584199250, 2806974975, -2058448620, 1091601510, -356440740, 100249155, -7426630, 1485041, 770, 45, -540, 90, 228, -105, -30, 35, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 + 1485041*x^10 - 7426630*x^9 + 100249155*x^8 - 356440740*x^7 + 1091601510*x^6 - 2058448620*x^5 + 2806974975*x^4 - 2584199250*x^3 + 1570569425*x^2 - 564365350*x + 551525305475)
 
gp: K = bnfinit(x^20 - 10*x^19 + 35*x^18 - 30*x^17 - 105*x^16 + 228*x^15 + 90*x^14 - 540*x^13 + 45*x^12 + 770*x^11 + 1485041*x^10 - 7426630*x^9 + 100249155*x^8 - 356440740*x^7 + 1091601510*x^6 - 2058448620*x^5 + 2806974975*x^4 - 2584199250*x^3 + 1570569425*x^2 - 564365350*x + 551525305475, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 35 x^{18} - 30 x^{17} - 105 x^{16} + 228 x^{15} + 90 x^{14} - 540 x^{13} + 45 x^{12} + 770 x^{11} + 1485041 x^{10} - 7426630 x^{9} + 100249155 x^{8} - 356440740 x^{7} + 1091601510 x^{6} - 2058448620 x^{5} + 2806974975 x^{4} - 2584199250 x^{3} + 1570569425 x^{2} - 564365350 x + 551525305475 \)

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:  \(90346941218160640000000000000000000000=2^{38}\cdot 5^{22}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.03$
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:  $2, 5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{13} a^{4} - \frac{2}{13} a^{3} - \frac{1}{13} a^{2} + \frac{2}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{5} - \frac{5}{13} a^{3} + \frac{5}{13} a + \frac{2}{13}$, $\frac{1}{13} a^{6} + \frac{3}{13} a^{3} - \frac{1}{13} a + \frac{5}{13}$, $\frac{1}{13} a^{7} + \frac{6}{13} a^{3} + \frac{2}{13} a^{2} - \frac{1}{13} a - \frac{3}{13}$, $\frac{1}{169} a^{8} - \frac{4}{169} a^{7} + \frac{2}{169} a^{6} - \frac{5}{169} a^{5} - \frac{5}{169} a^{4} + \frac{57}{169} a^{3} + \frac{2}{169} a^{2} - \frac{61}{169} a - \frac{25}{169}$, $\frac{1}{169} a^{9} - \frac{1}{169} a^{7} + \frac{3}{169} a^{6} + \frac{1}{169} a^{5} - \frac{2}{169} a^{4} - \frac{82}{169} a^{3} + \frac{12}{169} a^{2} - \frac{61}{169} a + \frac{43}{169}$, $\frac{1}{845} a^{10} + \frac{5}{169} a^{7} - \frac{2}{169} a^{6} + \frac{19}{845} a^{5} + \frac{6}{169} a^{4} + \frac{32}{169} a^{3} + \frac{9}{169} a^{2} - \frac{1}{169} a + \frac{34}{169}$, $\frac{1}{845} a^{11} + \frac{5}{169} a^{7} - \frac{31}{845} a^{6} + \frac{5}{169} a^{5} + \frac{5}{169} a^{4} + \frac{49}{169} a^{3} + \frac{15}{169} a^{2} - \frac{51}{169} a + \frac{60}{169}$, $\frac{1}{10985} a^{12} - \frac{6}{10985} a^{11} - \frac{4}{10985} a^{10} + \frac{2}{2197} a^{9} - \frac{6}{2197} a^{8} - \frac{331}{10985} a^{7} + \frac{171}{10985} a^{6} - \frac{241}{10985} a^{5} - \frac{84}{2197} a^{4} - \frac{418}{2197} a^{3} - \frac{994}{2197} a^{2} + \frac{893}{2197} a + \frac{437}{2197}$, $\frac{1}{10985} a^{13} - \frac{1}{10985} a^{11} - \frac{1}{10985} a^{10} + \frac{6}{2197} a^{9} + \frac{9}{10985} a^{8} - \frac{12}{2197} a^{7} - \frac{359}{10985} a^{6} + \frac{136}{10985} a^{5} + \frac{14}{2197} a^{4} - \frac{655}{2197} a^{3} + \frac{64}{2197} a^{2} + \frac{660}{2197} a + \frac{776}{2197}$, $\frac{1}{10985} a^{14} + \frac{6}{10985} a^{11} + \frac{19}{10985} a^{9} - \frac{5}{2197} a^{8} + \frac{83}{2197} a^{7} + \frac{294}{10985} a^{6} + \frac{36}{2197} a^{5} - \frac{50}{2197} a^{4} - \frac{315}{2197} a^{3} - \frac{516}{2197} a^{2} + \frac{239}{2197} a + \frac{177}{2197}$, $\frac{1}{10985} a^{15} - \frac{3}{10985} a^{11} + \frac{4}{10985} a^{10} - \frac{4}{2197} a^{9} + \frac{2}{2197} a^{8} + \frac{14}{2197} a^{7} - \frac{222}{10985} a^{6} - \frac{1}{169} a^{5} - \frac{19}{2197} a^{4} + \frac{55}{2197} a^{3} + \frac{119}{2197} a^{2} + \frac{487}{2197} a - \frac{438}{2197}$, $\frac{1}{714025} a^{16} - \frac{8}{714025} a^{15} - \frac{32}{714025} a^{14} - \frac{2}{54925} a^{13} - \frac{1}{142805} a^{12} + \frac{394}{714025} a^{11} - \frac{122}{714025} a^{10} + \frac{452}{714025} a^{9} + \frac{1526}{714025} a^{8} + \frac{5406}{142805} a^{7} + \frac{2508}{142805} a^{6} + \frac{836}{28561} a^{5} + \frac{5212}{142805} a^{4} + \frac{1351}{10985} a^{3} - \frac{6468}{28561} a^{2} + \frac{6219}{28561} a + \frac{4172}{28561}$, $\frac{1}{714025} a^{17} - \frac{31}{714025} a^{15} - \frac{22}{714025} a^{14} - \frac{18}{714025} a^{13} + \frac{29}{714025} a^{12} + \frac{47}{142805} a^{11} - \frac{4}{714025} a^{10} - \frac{1293}{714025} a^{9} - \frac{1582}{714025} a^{8} + \frac{5456}{142805} a^{7} + \frac{1728}{142805} a^{6} + \frac{479}{28561} a^{5} + \frac{4074}{142805} a^{4} - \frac{62591}{142805} a^{3} + \frac{6774}{28561} a^{2} + \frac{948}{2197} a - \frac{2322}{28561}$, $\frac{1}{329231944845781380491401097525} a^{18} - \frac{9}{329231944845781380491401097525} a^{17} + \frac{127165743541155887892128}{329231944845781380491401097525} a^{16} - \frac{203465189665849420627364}{65846388969156276098280219505} a^{15} - \frac{594382777810662396869016}{25325534218906260037800084425} a^{14} + \frac{11949944050718469194411762}{329231944845781380491401097525} a^{13} - \frac{7490578960556100092034761}{329231944845781380491401097525} a^{12} - \frac{96752036719170608942304668}{329231944845781380491401097525} a^{11} - \frac{11059921347978076713630697}{65846388969156276098280219505} a^{10} - \frac{548523900835540411900698027}{329231944845781380491401097525} a^{9} + \frac{633205760987868054740387572}{329231944845781380491401097525} a^{8} - \frac{122844454099007368326627209}{65846388969156276098280219505} a^{7} - \frac{1324291397580244979285631951}{65846388969156276098280219505} a^{6} + \frac{271438452379244222473772481}{65846388969156276098280219505} a^{5} - \frac{2519809250662604460270103789}{65846388969156276098280219505} a^{4} + \frac{16411144879508467995949227856}{65846388969156276098280219505} a^{3} - \frac{4515934402312351144956242463}{13169277793831255219656043901} a^{2} - \frac{5523087953924789546352015157}{13169277793831255219656043901} a + \frac{2477025156136554701952073933}{13169277793831255219656043901}$, $\frac{1}{250351654636963041173527712844890207225} a^{19} + \frac{76041117}{50070330927392608234705542568978041445} a^{18} - \frac{103165776990599755325490018724726}{250351654636963041173527712844890207225} a^{17} - \frac{126602647152771171701541045740017}{250351654636963041173527712844890207225} a^{16} + \frac{6009269162222154600663371941444227}{250351654636963041173527712844890207225} a^{15} - \frac{5860765696218792758512420833046136}{250351654636963041173527712844890207225} a^{14} - \frac{289683998522054936628275215320014}{10014066185478521646941108513795608289} a^{13} - \frac{1216964146039595887722278419975314}{250351654636963041173527712844890207225} a^{12} - \frac{64216059329574827144626840561322063}{250351654636963041173527712844890207225} a^{11} - \frac{23515883680027888215798384717074812}{250351654636963041173527712844890207225} a^{10} - \frac{5296728140126198823171486258895312}{3851563917491739094977349428382926265} a^{9} - \frac{27313781706263858843292662700510842}{50070330927392608234705542568978041445} a^{8} + \frac{413518955226252836040023797176975226}{50070330927392608234705542568978041445} a^{7} + \frac{1903863407037747003945041613430463672}{50070330927392608234705542568978041445} a^{6} - \frac{692912108915065637677960670855436104}{50070330927392608234705542568978041445} a^{5} + \frac{225618776339995647587992921775036062}{10014066185478521646941108513795608289} a^{4} - \frac{4573247632915341744598316422611289173}{10014066185478521646941108513795608289} a^{3} - \frac{3773058098659764485597610125386477223}{10014066185478521646941108513795608289} a^{2} + \frac{4870320798089471651012514746331918495}{10014066185478521646941108513795608289} a + \frac{3905317139599649762282886784173389598}{10014066185478521646941108513795608289}$

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

\(\Q(\sqrt{-26}) \), \(\Q(\sqrt{-130}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-26})\), 5.1.50000.1, 10.0.1901020160000000000.1, 10.0.9505100800000000000.1, 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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}$
$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}$