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Properties

Label 20.0.11008893815...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 401^{10}$
Root discriminant $89.55$
Ramified primes $2, 5, 401$
Class number $468512$ (GRH)
Class group $[11, 22, 22, 88]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![691769621, 101796464, 669533749, 106404156, 302499690, 49209408, 83640050, 13178894, 15627769, 2181498, 2048373, 210912, 187709, 7706, 11470, -678, 435, -96, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 14*x^18 - 96*x^17 + 435*x^16 - 678*x^15 + 11470*x^14 + 7706*x^13 + 187709*x^12 + 210912*x^11 + 2048373*x^10 + 2181498*x^9 + 15627769*x^8 + 13178894*x^7 + 83640050*x^6 + 49209408*x^5 + 302499690*x^4 + 106404156*x^3 + 669533749*x^2 + 101796464*x + 691769621)
 
gp: K = bnfinit(x^20 - 4*x^19 + 14*x^18 - 96*x^17 + 435*x^16 - 678*x^15 + 11470*x^14 + 7706*x^13 + 187709*x^12 + 210912*x^11 + 2048373*x^10 + 2181498*x^9 + 15627769*x^8 + 13178894*x^7 + 83640050*x^6 + 49209408*x^5 + 302499690*x^4 + 106404156*x^3 + 669533749*x^2 + 101796464*x + 691769621, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 14 x^{18} - 96 x^{17} + 435 x^{16} - 678 x^{15} + 11470 x^{14} + 7706 x^{13} + 187709 x^{12} + 210912 x^{11} + 2048373 x^{10} + 2181498 x^{9} + 15627769 x^{8} + 13178894 x^{7} + 83640050 x^{6} + 49209408 x^{5} + 302499690 x^{4} + 106404156 x^{3} + 669533749 x^{2} + 101796464 x + 691769621 \)

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:  \(1100889381588437644156968970240000000000=2^{20}\cdot 5^{10}\cdot 401^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.55$
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:  $2, 5, 401$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{6} + \frac{2}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{1}{9} a^{7} + \frac{2}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{27} a^{14} + \frac{1}{27} a^{13} + \frac{1}{27} a^{12} - \frac{4}{27} a^{11} - \frac{2}{27} a^{10} - \frac{1}{9} a^{9} + \frac{4}{27} a^{8} - \frac{2}{27} a^{7} + \frac{2}{27} a^{6} - \frac{4}{9} a^{5} - \frac{7}{27} a^{4} + \frac{11}{27} a^{3} + \frac{5}{27} a^{2} + \frac{1}{27} a - \frac{1}{27}$, $\frac{1}{27} a^{15} + \frac{1}{27} a^{12} + \frac{2}{27} a^{11} + \frac{2}{27} a^{10} - \frac{2}{27} a^{9} + \frac{1}{9} a^{8} + \frac{4}{27} a^{7} + \frac{1}{27} a^{6} + \frac{5}{27} a^{5} - \frac{2}{9} a^{3} + \frac{8}{27} a^{2} + \frac{7}{27} a + \frac{7}{27}$, $\frac{1}{81} a^{16} - \frac{1}{81} a^{14} - \frac{1}{27} a^{13} - \frac{2}{81} a^{12} - \frac{1}{9} a^{11} + \frac{4}{27} a^{10} - \frac{1}{27} a^{9} + \frac{1}{9} a^{6} + \frac{4}{27} a^{5} + \frac{10}{81} a^{4} + \frac{1}{9} a^{3} - \frac{4}{81} a^{2} + \frac{13}{27} a - \frac{20}{81}$, $\frac{1}{81} a^{17} - \frac{1}{81} a^{15} + \frac{1}{81} a^{13} + \frac{1}{27} a^{12} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{4}{27} a^{8} + \frac{1}{27} a^{7} - \frac{26}{81} a^{5} - \frac{13}{27} a^{4} + \frac{29}{81} a^{3} + \frac{2}{9} a^{2} - \frac{17}{81} a - \frac{7}{27}$, $\frac{1}{786591} a^{18} - \frac{43}{262197} a^{17} - \frac{319}{87399} a^{16} + \frac{170}{29133} a^{15} + \frac{508}{262197} a^{14} - \frac{1421}{29133} a^{13} + \frac{6784}{786591} a^{12} + \frac{7237}{87399} a^{11} - \frac{14290}{87399} a^{10} - \frac{37138}{262197} a^{9} - \frac{43273}{262197} a^{8} + \frac{2449}{29133} a^{7} - \frac{63761}{786591} a^{6} + \frac{22}{9711} a^{5} + \frac{41918}{87399} a^{4} - \frac{28189}{87399} a^{3} + \frac{9365}{20169} a^{2} - \frac{21940}{262197} a + \frac{268498}{786591}$, $\frac{1}{10081742361421649086650194749277221164412381121462076531} a^{19} - \frac{717218939265569758338499044818515549651307283695}{3360580787140549695550064916425740388137460373820692177} a^{18} + \frac{6525046458830961688072964973149848009882259685280135}{1120193595713516565183354972141913462712486791273564059} a^{17} + \frac{662163476073692125039323520212942581237022114328243}{1120193595713516565183354972141913462712486791273564059} a^{16} - \frac{52192700666388234374957789190649446189357187096066343}{3360580787140549695550064916425740388137460373820692177} a^{15} - \frac{156117221209739760806322813332155908517671952413940}{13496308382090561026305481592071246538704660135826073} a^{14} + \frac{272862885703891656737683350862500642601584762002380957}{10081742361421649086650194749277221164412381121462076531} a^{13} + \frac{90670749024109912437157345282079987886170494538365550}{3360580787140549695550064916425740388137460373820692177} a^{12} + \frac{3324145140332869494311036909134817871494082080062703}{86168738131808966552565767087839497131729753174889543} a^{11} - \frac{103275061173189010799410922316555838336477452082280}{3114532703559360236839726521247210739701075415959863} a^{10} + \frac{11069554184427096627843262953047139346173112446542312}{108405831843243538566131126336314206068950334639377167} a^{9} - \frac{156059115103206706283235324917284199151848235936478490}{1120193595713516565183354972141913462712486791273564059} a^{8} - \frac{166933763904810684919699636449464135088481842464896994}{10081742361421649086650194749277221164412381121462076531} a^{7} - \frac{198019488396621029982393078472690122187141676835600415}{3360580787140549695550064916425740388137460373820692177} a^{6} - \frac{95739745261802044882152519203702430928293943184174600}{373397865237838855061118324047304487570828930424521353} a^{5} - \frac{51374005853396067454248970761999580132136619502769771}{373397865237838855061118324047304487570828930424521353} a^{4} + \frac{881302344813566762963949606074652962666072244054105254}{3360580787140549695550064916425740388137460373820692177} a^{3} - \frac{712967277835235342788936908351529183451779715793397303}{3360580787140549695550064916425740388137460373820692177} a^{2} + \frac{162855725536971382577971122551543411367349521016410142}{775518643186280698973091903790555474185567778574005887} a + \frac{22942434870578223489649793531903754938513261279150068}{63407184663029239538680470121240384681838874977748909}$

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

Class group and class number

$C_{11}\times C_{22}\times C_{22}\times C_{88}$, which has order $468512$ (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:  \( 795087.603907 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{401}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-2005}) \), \(\Q(\sqrt{-5}, \sqrt{401})\), 5.5.160801.1 x5, 10.10.10368641602001.1, 10.0.82742277123200000.1 x5, 10.0.33179653126403200000.2 x5

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

Sibling fields

Degree 10 siblings: 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
401Data not computed