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Properties

Label 20.0.95012027507...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 19^{10}$
Root discriminant $62.93$
Ramified primes $2, 3, 5, 19$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4194304, 0, 11534336, 5898240, 14401536, 7981056, 9604608, 4041216, 3174528, 929592, 637014, 111264, 84600, -1932, 9846, -1557, 885, -132, 32, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 32*x^18 - 132*x^17 + 885*x^16 - 1557*x^15 + 9846*x^14 - 1932*x^13 + 84600*x^12 + 111264*x^11 + 637014*x^10 + 929592*x^9 + 3174528*x^8 + 4041216*x^7 + 9604608*x^6 + 7981056*x^5 + 14401536*x^4 + 5898240*x^3 + 11534336*x^2 + 4194304)
 
gp: K = bnfinit(x^20 - 3*x^19 + 32*x^18 - 132*x^17 + 885*x^16 - 1557*x^15 + 9846*x^14 - 1932*x^13 + 84600*x^12 + 111264*x^11 + 637014*x^10 + 929592*x^9 + 3174528*x^8 + 4041216*x^7 + 9604608*x^6 + 7981056*x^5 + 14401536*x^4 + 5898240*x^3 + 11534336*x^2 + 4194304, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 32 x^{18} - 132 x^{17} + 885 x^{16} - 1557 x^{15} + 9846 x^{14} - 1932 x^{13} + 84600 x^{12} + 111264 x^{11} + 637014 x^{10} + 929592 x^{9} + 3174528 x^{8} + 4041216 x^{7} + 9604608 x^{6} + 7981056 x^{5} + 14401536 x^{4} + 5898240 x^{3} + 11534336 x^{2} + 4194304 \)

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:  \(950120275075465393875600000000000000=2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.93$
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:  $2, 3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{3}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{11} + \frac{1}{8} a^{10} + \frac{3}{8} a^{9} + \frac{5}{32} a^{8} - \frac{1}{32} a^{7} - \frac{7}{16} a^{6} - \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{5}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{13} + \frac{1}{128} a^{12} + \frac{1}{32} a^{11} - \frac{5}{32} a^{10} + \frac{37}{128} a^{9} - \frac{1}{128} a^{8} - \frac{23}{64} a^{7} - \frac{9}{32} a^{6} + \frac{5}{16} a^{5} - \frac{1}{2} a^{4} - \frac{21}{64} a^{3} + \frac{1}{8} a^{2}$, $\frac{1}{512} a^{14} + \frac{1}{512} a^{13} + \frac{1}{128} a^{12} - \frac{5}{128} a^{11} + \frac{37}{512} a^{10} - \frac{1}{512} a^{9} - \frac{23}{256} a^{8} + \frac{23}{128} a^{7} + \frac{5}{64} a^{6} - \frac{1}{8} a^{5} + \frac{43}{256} a^{4} + \frac{9}{32} a^{3} - \frac{1}{2} a$, $\frac{1}{2048} a^{15} + \frac{1}{2048} a^{14} + \frac{1}{512} a^{13} - \frac{5}{512} a^{12} + \frac{37}{2048} a^{11} - \frac{1}{2048} a^{10} - \frac{23}{1024} a^{9} - \frac{105}{512} a^{8} + \frac{69}{256} a^{7} - \frac{1}{32} a^{6} - \frac{213}{1024} a^{5} + \frac{9}{128} a^{4} + \frac{3}{8} a^{2}$, $\frac{1}{40960} a^{16} + \frac{9}{40960} a^{15} - \frac{1}{10240} a^{14} - \frac{17}{10240} a^{13} + \frac{261}{40960} a^{12} + \frac{871}{40960} a^{11} - \frac{4803}{20480} a^{10} + \frac{2287}{10240} a^{9} + \frac{1349}{5120} a^{8} - \frac{237}{640} a^{7} + \frac{1199}{4096} a^{6} - \frac{267}{640} a^{5} - \frac{7}{1280} a^{4} - \frac{119}{320} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{163840} a^{17} + \frac{1}{163840} a^{16} + \frac{1}{40960} a^{15} + \frac{11}{40960} a^{14} - \frac{31}{32768} a^{13} + \frac{1023}{163840} a^{12} + \frac{3433}{81920} a^{11} - \frac{2841}{40960} a^{10} - \frac{6499}{20480} a^{9} + \frac{869}{2560} a^{8} - \frac{17493}{81920} a^{7} - \frac{23}{10240} a^{6} - \frac{5}{16} a^{5} - \frac{67}{256} a^{4} + \frac{151}{320} a^{3} - \frac{37}{80} a^{2} + \frac{9}{20} a + \frac{1}{5}$, $\frac{1}{683540480} a^{18} + \frac{281}{683540480} a^{17} - \frac{1829}{170885120} a^{16} - \frac{33449}{170885120} a^{15} - \frac{18623}{136708096} a^{14} + \frac{871223}{683540480} a^{13} + \frac{4758693}{341770240} a^{12} + \frac{7272159}{170885120} a^{11} - \frac{16142859}{85442560} a^{10} + \frac{4435439}{10680320} a^{9} + \frac{9681661}{48824320} a^{8} + \frac{4772741}{21360640} a^{7} - \frac{1625465}{4272128} a^{6} - \frac{403019}{1068032} a^{5} - \frac{9411}{20860} a^{4} + \frac{18089}{47680} a^{3} + \frac{5149}{83440} a^{2} + \frac{1311}{20860} a + \frac{419}{1043}$, $\frac{1}{11676121160773778887475318687759151693695549440} a^{19} + \frac{3563330995177535127637373450101461013}{11676121160773778887475318687759151693695549440} a^{18} - \frac{72613786242604289958962737105027975799}{1459515145096722360934414835969893961711943680} a^{17} + \frac{4400068137742075946739555201093284843421}{417004327170492103124118524562826846203412480} a^{16} - \frac{2415506118869181766100942664869639920015083}{11676121160773778887475318687759151693695549440} a^{15} + \frac{1277362573854255899525258239588728559988487}{2335224232154755777495063737551830338739109888} a^{14} + \frac{3332049370114506105887346052189099394308907}{1167612116077377888747531868775915169369554944} a^{13} - \frac{24455736490466681738468056656958186057837163}{2919030290193444721868829671939787923423887360} a^{12} + \frac{40289039595031657706351101764773040899652767}{1459515145096722360934414835969893961711943680} a^{11} - \frac{6195423988808238213240901531493071154590847}{72975757254836118046720741798494698085597184} a^{10} + \frac{1919293884110095256208414603707122967597636587}{5838060580386889443737659343879575846847774720} a^{9} + \frac{3763389855505523671837141178719645172047189}{291903029019344472186882967193978792342388736} a^{8} - \frac{28746020873703720511194877355571071980007329}{72975757254836118046720741798494698085597184} a^{7} + \frac{43515780828369418830828737519087789709133213}{91219696568545147558400927248118372606996480} a^{6} - \frac{1883844194685965569534873249360924799918717}{4560984828427257377920046362405918630349824} a^{5} + \frac{460376567035738772503657276837865886598791}{1425307758883517930600014488251849571984320} a^{4} - \frac{7401151620368941967343785938468376314081}{22270433732554967665625226378935149562255} a^{3} + \frac{55156546366685486205041579262454957665127}{356326939720879482650003622062962392996080} a^{2} - \frac{3211466966034383491100294429620799847671}{8908173493021987066250090551574059824902} a - \frac{10737231487443694954986725200423204700261}{22270433732554967665625226378935149562255}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 16331936688.22532 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{57}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-95}) \), \(\Q(\sqrt{-15}, \sqrt{57})\), 5.1.162000.1, 10.2.194948226468000000.1, 10.0.393660000000.1, 10.0.324913710780000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$