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Properties

Label 20.0.47225249792...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{34}\cdot 5^{21}\cdot 7^{8}$
Root discriminant $38.34$
Ramified primes $2, 5, 7$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_2^2:F_5$ (as 20T19)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, 80, 0, -160, 9200, 2024, -18720, -800, 13600, -1800, -3960, 960, 1100, -140, -400, 0, 60, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 60*x^16 - 400*x^14 - 140*x^13 + 1100*x^12 + 960*x^11 - 3960*x^10 - 1800*x^9 + 13600*x^8 - 800*x^7 - 18720*x^6 + 2024*x^5 + 9200*x^4 - 160*x^3 + 80*x + 8)
 
gp: K = bnfinit(x^20 + 60*x^16 - 400*x^14 - 140*x^13 + 1100*x^12 + 960*x^11 - 3960*x^10 - 1800*x^9 + 13600*x^8 - 800*x^7 - 18720*x^6 + 2024*x^5 + 9200*x^4 - 160*x^3 + 80*x + 8, 1)
 

Normalized defining polynomial

\( x^{20} + 60 x^{16} - 400 x^{14} - 140 x^{13} + 1100 x^{12} + 960 x^{11} - 3960 x^{10} - 1800 x^{9} + 13600 x^{8} - 800 x^{7} - 18720 x^{6} + 2024 x^{5} + 9200 x^{4} - 160 x^{3} + 80 x + 8 \)

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:  \(47225249792000000000000000000000=2^{34}\cdot 5^{21}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.34$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{4} a^{14}$, $\frac{1}{20} a^{15} + \frac{1}{5} a^{10} - \frac{1}{5} a^{5} + \frac{2}{5}$, $\frac{1}{20} a^{16} + \frac{1}{5} a^{11} - \frac{1}{5} a^{6} + \frac{2}{5} a$, $\frac{1}{60} a^{17} - \frac{1}{12} a^{14} - \frac{1}{10} a^{12} + \frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{7}{30} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{5} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{900} a^{18} - \frac{7}{900} a^{17} + \frac{1}{300} a^{16} - \frac{7}{450} a^{15} + \frac{1}{45} a^{14} - \frac{6}{25} a^{13} + \frac{16}{75} a^{12} - \frac{49}{450} a^{11} - \frac{103}{450} a^{10} + \frac{8}{45} a^{9} + \frac{83}{450} a^{8} + \frac{7}{225} a^{7} + \frac{92}{225} a^{6} + \frac{49}{225} a^{5} - \frac{2}{15} a^{4} + \frac{82}{225} a^{3} - \frac{44}{225} a^{2} + \frac{32}{75} a + \frac{47}{225}$, $\frac{1}{2007681946359438643853210048700} a^{19} - \frac{380003398265905393164470473}{1003840973179719321926605024350} a^{18} + \frac{2124312680812390812881411963}{1003840973179719321926605024350} a^{17} - \frac{7777602651742404046156605674}{501920486589859660963302512175} a^{16} + \frac{3131244481953651480606659629}{501920486589859660963302512175} a^{15} + \frac{15263594303743510847166425476}{501920486589859660963302512175} a^{14} - \frac{80968874434477258038407805139}{334613657726573107308868341450} a^{13} - \frac{7808164127692167469009348243}{1003840973179719321926605024350} a^{12} + \frac{16112788967927930979710517767}{111537885908857702436289447150} a^{11} + \frac{823099441527626051366056429}{10793988958921713138995752950} a^{10} - \frac{76334576871850140969935053079}{334613657726573107308868341450} a^{9} + \frac{37371845358417640618229347142}{167306828863286553654434170725} a^{8} + \frac{28825985249309214071315733244}{501920486589859660963302512175} a^{7} - \frac{147271003701595147157537844124}{501920486589859660963302512175} a^{6} - \frac{137856925837622237146593753391}{501920486589859660963302512175} a^{5} + \frac{129468749465750431627235439652}{501920486589859660963302512175} a^{4} + \frac{13891581813980317566120785987}{55768942954428851218144723575} a^{3} - \frac{123119493259295266380069208663}{501920486589859660963302512175} a^{2} - \frac{136261931440972408489086701152}{501920486589859660963302512175} a - \frac{84950622174759935521443361283}{501920486589859660963302512175}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( -\frac{39013913531359184525}{715461855720418299978} a^{19} + \frac{1446073488390607315}{238487285240139433326} a^{18} - \frac{403756309838218550}{357730927860209149989} a^{17} + \frac{215538187451108840}{357730927860209149989} a^{16} - \frac{780265508997704197381}{238487285240139433326} a^{15} + \frac{130361888598309392650}{357730927860209149989} a^{14} + \frac{5185749820332637798285}{238487285240139433326} a^{13} + \frac{1876525831743628034900}{357730927860209149989} a^{12} - \frac{21598555216433323090355}{357730927860209149989} a^{11} - \frac{1055292243189015205399}{23079414700658654838} a^{10} + \frac{78852201996013925263375}{357730927860209149989} a^{9} + \frac{26300516271751297431250}{357730927860209149989} a^{8} - \frac{89140466809531119330250}{119243642620069716663} a^{7} + \frac{15048798347392974779315}{119243642620069716663} a^{6} + \frac{357690717580880624813783}{357730927860209149989} a^{5} - \frac{77463478877276102562100}{357730927860209149989} a^{4} - \frac{167803508152543425660830}{357730927860209149989} a^{3} + \frac{20234944650223835014550}{357730927860209149989} a^{2} - \frac{3832828609403781660110}{357730927860209149989} a - \frac{1138629933805061206727}{357730927860209149989} \) (order $4$)
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:  \( 32474897.1206 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:F_5$ (as 20T19):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_2^2:F_5$
Character table for $C_2^2:F_5$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.320.1, 5.5.2450000.1, 10.0.384160000000000.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$5.10.11.8$x^{10} + 20 x^{2} + 10$$10$$1$$11$$F_{5}\times C_2$$[5/4]_{4}^{2}$
5.10.10.10$x^{10} + 10 x^{8} + 5 x^{6} + 10 x^{5} - 20 x^{4} - 20 x^{2} + 2$$5$$2$$10$$F_{5}\times C_2$$[5/4]_{4}^{2}$
$7$7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$