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Properties

Label 20.0.41528275278...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 5^{22}\cdot 7^{10}\cdot 97^{2}$
Root discriminant $42.75$
Ramified primes $2, 5, 7, 97$
Class number $156$ (GRH)
Class group $[156]$ (GRH)
Galois group 20T140

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![162871, 483750, 629335, 442940, 171825, 42730, 68790, 82980, 29145, -3050, 1117, 3020, 655, 200, 90, -60, -15, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 15*x^16 - 60*x^15 + 90*x^14 + 200*x^13 + 655*x^12 + 3020*x^11 + 1117*x^10 - 3050*x^9 + 29145*x^8 + 82980*x^7 + 68790*x^6 + 42730*x^5 + 171825*x^4 + 442940*x^3 + 629335*x^2 + 483750*x + 162871)
 
gp: K = bnfinit(x^20 - 5*x^18 - 15*x^16 - 60*x^15 + 90*x^14 + 200*x^13 + 655*x^12 + 3020*x^11 + 1117*x^10 - 3050*x^9 + 29145*x^8 + 82980*x^7 + 68790*x^6 + 42730*x^5 + 171825*x^4 + 442940*x^3 + 629335*x^2 + 483750*x + 162871, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{18} - 15 x^{16} - 60 x^{15} + 90 x^{14} + 200 x^{13} + 655 x^{12} + 3020 x^{11} + 1117 x^{10} - 3050 x^{9} + 29145 x^{8} + 82980 x^{7} + 68790 x^{6} + 42730 x^{5} + 171825 x^{4} + 442940 x^{3} + 629335 x^{2} + 483750 x + 162871 \)

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:  \(415282752787656250000000000000000=2^{16}\cdot 5^{22}\cdot 7^{10}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.75$
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:  $2, 5, 7, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{14} a^{18} - \frac{3}{14} a^{17} - \frac{1}{7} a^{15} - \frac{1}{7} a^{14} + \frac{3}{14} a^{13} - \frac{1}{7} a^{12} - \frac{1}{7} a^{11} - \frac{3}{14} a^{10} - \frac{1}{14} a^{9} + \frac{5}{14} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} - \frac{1}{7} a^{5} + \frac{5}{14} a^{4} + \frac{1}{7} a^{3} + \frac{5}{14} a^{2} + \frac{3}{7} a - \frac{3}{14}$, $\frac{1}{5944856333908765191042608221874175267973914870526} a^{19} + \frac{37910086516633804245994049922747714930433204649}{5944856333908765191042608221874175267973914870526} a^{18} + \frac{471938079815318763979592819794164619996664671833}{5944856333908765191042608221874175267973914870526} a^{17} + \frac{291213385547454470932457405924952859776832104976}{2972428166954382595521304110937087633986957435263} a^{16} + \frac{732280346358663321008234359200011354541679073925}{2972428166954382595521304110937087633986957435263} a^{15} - \frac{256389240684891908905655639383887008246098027579}{5944856333908765191042608221874175267973914870526} a^{14} + \frac{124214321103188527095985061714423717628075420923}{5944856333908765191042608221874175267973914870526} a^{13} + \frac{721595473371200040053627310091479785036926783513}{5944856333908765191042608221874175267973914870526} a^{12} - \frac{226216972924845812008660552341365415365153834326}{2972428166954382595521304110937087633986957435263} a^{11} + \frac{233339350149537063934553826670051237093781648898}{2972428166954382595521304110937087633986957435263} a^{10} + \frac{121631998106110616360115258705659855692629677859}{2972428166954382595521304110937087633986957435263} a^{9} + \frac{1447344540797952246064793227791607285369391947017}{2972428166954382595521304110937087633986957435263} a^{8} - \frac{42752228585573978671780078832325576573939234583}{5944856333908765191042608221874175267973914870526} a^{7} + \frac{346338108419063835142881134410027028886699684131}{849265190558395027291801174553453609710559267218} a^{6} - \frac{1112994231722550603681952411273201436501895101855}{5944856333908765191042608221874175267973914870526} a^{5} - \frac{70237661030829316707770096130164776268609761873}{2972428166954382595521304110937087633986957435263} a^{4} - \frac{701692836830439205053237183508617428921920326134}{2972428166954382595521304110937087633986957435263} a^{3} - \frac{633172811723040644466340788198916221104888111503}{5944856333908765191042608221874175267973914870526} a^{2} - \frac{116705134768730296614844303261835360424127792493}{849265190558395027291801174553453609710559267218} a - \frac{1589837644253478764767986349042418543686512292101}{5944856333908765191042608221874175267973914870526}$

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

Class group and class number

$C_{156}$, which has order $156$ (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:  \( 1633832.53547 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T140:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 22 conjugacy class representatives for t20n140
Character table for t20n140 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.2450000.1, 10.0.4075697500000000.1, 10.0.20378487500000000.1, 10.10.30012500000000.1

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
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R R ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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}$
$5$5.10.11.1$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.1$x^{10} + 20 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$97$97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$
97.4.0.1$x^{4} - x + 23$$1$$4$$0$$C_4$$[\ ]^{4}$