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Properties

Label 20.0.18600378723...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 17^{10}$
Root discriminant $73.03$
Ramified primes $2, 3, 5, 17$
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![62307524980, -670653100, 57642488220, -238929480, 23801041500, -7525230, 5688096720, 4713030, 866646000, 594140, 87898049, 24610, 6015205, 230, 274580, -4, 8010, 0, 135, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 135*x^18 + 8010*x^16 - 4*x^15 + 274580*x^14 + 230*x^13 + 6015205*x^12 + 24610*x^11 + 87898049*x^10 + 594140*x^9 + 866646000*x^8 + 4713030*x^7 + 5688096720*x^6 - 7525230*x^5 + 23801041500*x^4 - 238929480*x^3 + 57642488220*x^2 - 670653100*x + 62307524980)
 
gp: K = bnfinit(x^20 + 135*x^18 + 8010*x^16 - 4*x^15 + 274580*x^14 + 230*x^13 + 6015205*x^12 + 24610*x^11 + 87898049*x^10 + 594140*x^9 + 866646000*x^8 + 4713030*x^7 + 5688096720*x^6 - 7525230*x^5 + 23801041500*x^4 - 238929480*x^3 + 57642488220*x^2 - 670653100*x + 62307524980, 1)
 

Normalized defining polynomial

\( x^{20} + 135 x^{18} + 8010 x^{16} - 4 x^{15} + 274580 x^{14} + 230 x^{13} + 6015205 x^{12} + 24610 x^{11} + 87898049 x^{10} + 594140 x^{9} + 866646000 x^{8} + 4713030 x^{7} + 5688096720 x^{6} - 7525230 x^{5} + 23801041500 x^{4} - 238929480 x^{3} + 57642488220 x^{2} - 670653100 x + 62307524980 \)

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:  \(18600378723064531406250000000000000000=2^{16}\cdot 3^{10}\cdot 5^{22}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.03$
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:  $2, 3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9}$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} + \frac{3}{14} a^{16} + \frac{1}{14} a^{15} + \frac{3}{14} a^{14} + \frac{1}{7} a^{12} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} + \frac{5}{14} a^{8} + \frac{3}{14} a^{7} - \frac{2}{7} a^{6} - \frac{5}{14} a^{5} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2} + \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{208319165211426994988860802416255678150665271787999324382418121190559302586} a^{19} - \frac{28855448331508977902260220487619462331724331526046225459461807256702765}{29759880744489570712694400345179382592952181683999903483202588741508471798} a^{18} + \frac{3726927180152522751347990754250058738988195758650985046837386854043552233}{208319165211426994988860802416255678150665271787999324382418121190559302586} a^{17} - \frac{24062349815545617179602305576410455760530001908350383203420709057703744534}{104159582605713497494430401208127839075332635893999662191209060595279651293} a^{16} - \frac{708267371958770582163343903911395478203859609684231068090606839351940952}{104159582605713497494430401208127839075332635893999662191209060595279651293} a^{15} - \frac{10948236333696377070827110962995363838535777743458293852161026231678802163}{104159582605713497494430401208127839075332635893999662191209060595279651293} a^{14} + \frac{22005968927328137337945987023661815983369503388064227714057733109911703953}{104159582605713497494430401208127839075332635893999662191209060595279651293} a^{13} + \frac{14883532247583998947951197096206879555965174007130214597594878615152450687}{208319165211426994988860802416255678150665271787999324382418121190559302586} a^{12} + \frac{12744143254057603468343020224606716656227555067291355245758474902655375692}{104159582605713497494430401208127839075332635893999662191209060595279651293} a^{11} - \frac{28448520725796749723404751696124522992247962849769398852173533786695481709}{208319165211426994988860802416255678150665271787999324382418121190559302586} a^{10} - \frac{24126195818767789326157172678062133339833157313932924479555387628332367109}{208319165211426994988860802416255678150665271787999324382418121190559302586} a^{9} - \frac{56160532683935891659380601734146871962060608381066888552039990531668580315}{208319165211426994988860802416255678150665271787999324382418121190559302586} a^{8} + \frac{56482041325478397963168531415476747551034701659174039068245961730886653567}{208319165211426994988860802416255678150665271787999324382418121190559302586} a^{7} - \frac{40586073566325691909187467987025630768490646182310762711703129785337274883}{104159582605713497494430401208127839075332635893999662191209060595279651293} a^{6} + \frac{4158521012208506210611840044632393223456679920910363188028015032835683916}{104159582605713497494430401208127839075332635893999662191209060595279651293} a^{5} - \frac{41499785525054160577310337625821744759103634117181652464316990852873720525}{104159582605713497494430401208127839075332635893999662191209060595279651293} a^{4} + \frac{5081029255219214514990512897226157432009698952422409002749631328104673906}{104159582605713497494430401208127839075332635893999662191209060595279651293} a^{3} - \frac{23691493969712610484068057607047336305394355785782976754164634694852937387}{104159582605713497494430401208127839075332635893999662191209060595279651293} a^{2} + \frac{35609838742786875512226769426956176797960609768147654242497422788478990867}{104159582605713497494430401208127839075332635893999662191209060595279651293} a + \frac{26850694403111700453169475725597818193411528400151850113804643220245759533}{104159582605713497494430401208127839075332635893999662191209060595279651293}$

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

\(\Q(\sqrt{-255}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{-51})\), 5.1.50000.1, 10.0.4312815637500000000.1, 10.0.862563127500000000.1, 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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.8.1$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$$[5/4]_{4}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$