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Properties

Label 20.8.61257625575...0496.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{48}\cdot 31^{10}\cdot 227^{4}$
Root discriminant $86.97$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-369536, 1273344, 12280096, -732960, -34808130, 13096568, 1887716, 2160304, 3892137, -823644, 1185192, -331524, 217183, -60828, 22066, -5704, 1343, -316, 42, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 42*x^18 - 316*x^17 + 1343*x^16 - 5704*x^15 + 22066*x^14 - 60828*x^13 + 217183*x^12 - 331524*x^11 + 1185192*x^10 - 823644*x^9 + 3892137*x^8 + 2160304*x^7 + 1887716*x^6 + 13096568*x^5 - 34808130*x^4 - 732960*x^3 + 12280096*x^2 + 1273344*x - 369536)
 
gp: K = bnfinit(x^20 - 8*x^19 + 42*x^18 - 316*x^17 + 1343*x^16 - 5704*x^15 + 22066*x^14 - 60828*x^13 + 217183*x^12 - 331524*x^11 + 1185192*x^10 - 823644*x^9 + 3892137*x^8 + 2160304*x^7 + 1887716*x^6 + 13096568*x^5 - 34808130*x^4 - 732960*x^3 + 12280096*x^2 + 1273344*x - 369536, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 42 x^{18} - 316 x^{17} + 1343 x^{16} - 5704 x^{15} + 22066 x^{14} - 60828 x^{13} + 217183 x^{12} - 331524 x^{11} + 1185192 x^{10} - 823644 x^{9} + 3892137 x^{8} + 2160304 x^{7} + 1887716 x^{6} + 13096568 x^{5} - 34808130 x^{4} - 732960 x^{3} + 12280096 x^{2} + 1273344 x - 369536 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(612576255759573370734751264818904170496=2^{48}\cdot 31^{10}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.97$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{17} - \frac{1}{2} a^{15} - \frac{1}{4} a^{13} - \frac{1}{2} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{48} a^{18} - \frac{1}{12} a^{17} - \frac{1}{8} a^{16} - \frac{5}{12} a^{15} + \frac{5}{16} a^{14} - \frac{1}{4} a^{13} - \frac{7}{24} a^{12} - \frac{5}{12} a^{11} + \frac{5}{16} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{5}{12} a^{7} + \frac{3}{16} a^{6} - \frac{1}{4} a^{5} + \frac{5}{12} a^{4} - \frac{1}{6} a^{3} - \frac{3}{8} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{9505782975930384944858876083764697643204350118968541757582221995339552918848} a^{19} + \frac{2604031130349112566103549032068578853460673481692996954088716471950375}{2244046972599241016255636469255122200945314003533650084415066571137760368} a^{18} + \frac{558799042084016379758135652941468390451623810810180100331801487779762178065}{4752891487965192472429438041882348821602175059484270878791110997669776459424} a^{17} - \frac{94854104605140584401148865688285043877589006437460460949114367836738720809}{2376445743982596236214719020941174410801087529742135439395555498834888229712} a^{16} + \frac{1856259196591628895083798425165410545958221350876797119484653809672397228959}{9505782975930384944858876083764697643204350118968541757582221995339552918848} a^{15} + \frac{44619949023689159033761829646317058335235314864428301178324944712813290597}{792148581327532078738239673647058136933695843247378479798518499611629409904} a^{14} + \frac{1205045671684344419532597307588931435121652965087283246659800654227548812669}{4752891487965192472429438041882348821602175059484270878791110997669776459424} a^{13} - \frac{723984109763274261387651726807428568891378132464803582892713605900697053529}{2376445743982596236214719020941174410801087529742135439395555498834888229712} a^{12} + \frac{1556474321814526194312419026203708789993513337614760765512133907142519272383}{9505782975930384944858876083764697643204350118968541757582221995339552918848} a^{11} - \frac{37416345868332583354635643505840776011651882999642722033640964395709026101}{148527858998912264763419938808823400675067970608883464962222218677180514357} a^{10} + \frac{107706007155097456396264723452722893743311952669763520267140505392267672147}{297055717997824529526839877617646801350135941217766929924444437354361028714} a^{9} + \frac{103049796041043563441228100868252626688593719020015308144899024154103557107}{792148581327532078738239673647058136933695843247378479798518499611629409904} a^{8} + \frac{2454606358347353926383482270603710848403062468269296321869277926740074072601}{9505782975930384944858876083764697643204350118968541757582221995339552918848} a^{7} - \frac{204974641042072601341068695997635477778272143873657147317074831411796108095}{792148581327532078738239673647058136933695843247378479798518499611629409904} a^{6} - \frac{890613737897696031709105605597786537706773821503231014875714115854083739377}{2376445743982596236214719020941174410801087529742135439395555498834888229712} a^{5} - \frac{67915474160410858139068551091314279567372619127069420812929510059752751577}{396074290663766039369119836823529068466847921623689239899259249805814704952} a^{4} - \frac{665608633453591718157128260993826620449867317718847837840949048834594550625}{4752891487965192472429438041882348821602175059484270878791110997669776459424} a^{3} - \frac{557189091409708016240685309237826353041057988772584068300531179736783337979}{1188222871991298118107359510470587205400543764871067719697777749417444114856} a^{2} + \frac{31904087836888995185851298066337240175278737136317965927696107162255992117}{198037145331883019684559918411764534233423960811844619949629624902907352476} a + \frac{23029646553254736225950801357504550614445779397813437916422812308495415826}{148527858998912264763419938808823400675067970608883464962222218677180514357}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 4894419339420 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.2$x^{4} + 8 x + 14$$4$$1$$11$$C_4$$[3, 4]$
2.12.26.27$x^{12} - 2 x^{10} + 4 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{3} + 2$$12$$1$$26$12T48$[4/3, 4/3, 2, 3]_{3}^{2}$
31Data not computed
227Data not computed