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Properties

Label 20.0.54521108604...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 3^{17}\cdot 5^{20}\cdot 11^{5}$
Root discriminant $86.46$
Ramified primes $2, 3, 5, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![71696, -162240, 191840, -267040, 312880, -200888, 89320, -98360, 127410, -83440, 29993, -1890, -2335, 640, 130, -284, 220, -120, 45, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 45*x^18 - 120*x^17 + 220*x^16 - 284*x^15 + 130*x^14 + 640*x^13 - 2335*x^12 - 1890*x^11 + 29993*x^10 - 83440*x^9 + 127410*x^8 - 98360*x^7 + 89320*x^6 - 200888*x^5 + 312880*x^4 - 267040*x^3 + 191840*x^2 - 162240*x + 71696)
 
gp: K = bnfinit(x^20 - 10*x^19 + 45*x^18 - 120*x^17 + 220*x^16 - 284*x^15 + 130*x^14 + 640*x^13 - 2335*x^12 - 1890*x^11 + 29993*x^10 - 83440*x^9 + 127410*x^8 - 98360*x^7 + 89320*x^6 - 200888*x^5 + 312880*x^4 - 267040*x^3 + 191840*x^2 - 162240*x + 71696, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 45 x^{18} - 120 x^{17} + 220 x^{16} - 284 x^{15} + 130 x^{14} + 640 x^{13} - 2335 x^{12} - 1890 x^{11} + 29993 x^{10} - 83440 x^{9} + 127410 x^{8} - 98360 x^{7} + 89320 x^{6} - 200888 x^{5} + 312880 x^{4} - 267040 x^{3} + 191840 x^{2} - 162240 x + 71696 \)

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:  \(545211086046835507200000000000000000000=2^{38}\cdot 3^{17}\cdot 5^{20}\cdot 11^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.46$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6}$, $\frac{1}{12} a^{15} - \frac{1}{12} a^{13} + \frac{1}{12} a^{12} + \frac{1}{12} a^{11} + \frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} + \frac{1}{6} a^{7} - \frac{1}{4} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{24} a^{16} + \frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} + \frac{1}{12} a^{10} + \frac{5}{24} a^{8} - \frac{1}{4} a^{7} + \frac{1}{6} a^{6} + \frac{1}{12} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{24} a^{17} - \frac{1}{12} a^{14} + \frac{1}{12} a^{12} - \frac{1}{12} a^{10} + \frac{1}{24} a^{9} - \frac{1}{4} a^{8} - \frac{1}{6} a^{6} - \frac{1}{12} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{567243672658296} a^{18} + \frac{1123002601925}{141810918164574} a^{17} + \frac{1189326562948}{70905459082287} a^{16} - \frac{2965318345997}{141810918164574} a^{15} + \frac{1542674702971}{47270306054858} a^{14} - \frac{33433403664499}{283621836329148} a^{13} - \frac{6933356736565}{94540612109716} a^{12} + \frac{1647767103764}{23635153027429} a^{11} - \frac{48489370833227}{567243672658296} a^{10} - \frac{30521917596341}{283621836329148} a^{9} - \frac{25406176882267}{283621836329148} a^{8} + \frac{9724749245725}{70905459082287} a^{7} + \frac{27516604332275}{283621836329148} a^{6} - \frac{44397204093833}{141810918164574} a^{5} + \frac{54046621385041}{141810918164574} a^{4} - \frac{1727341552187}{47270306054858} a^{3} - \frac{25723090792451}{70905459082287} a^{2} + \frac{31859295016904}{70905459082287} a + \frac{18090761738213}{70905459082287}$, $\frac{1}{8686154035600365568268583672552} a^{19} + \frac{2168432030954503}{8686154035600365568268583672552} a^{18} - \frac{39841915649676811822789185505}{2171538508900091392067145918138} a^{17} + \frac{25139498918663161870538821141}{8686154035600365568268583672552} a^{16} - \frac{4474071786754342102583739508}{361923084816681898677857653023} a^{15} + \frac{378596101604781165065639057309}{4343077017800182784134291836276} a^{14} - \frac{25483493144039908426149374707}{482564113088909198237143537364} a^{13} + \frac{217955953660749006057403270417}{2171538508900091392067145918138} a^{12} + \frac{25130992369919723500649941255}{8686154035600365568268583672552} a^{11} - \frac{338242286516375661797980366615}{8686154035600365568268583672552} a^{10} - \frac{287405267830579288393976905307}{1447692339266727594711430612092} a^{9} - \frac{1219919384546036903628359078377}{8686154035600365568268583672552} a^{8} + \frac{472794014195141913878171632933}{2171538508900091392067145918138} a^{7} + \frac{234105572560005303436420837945}{1447692339266727594711430612092} a^{6} + \frac{169578775071912134810715243391}{2171538508900091392067145918138} a^{5} - \frac{85577767539085055328076628362}{361923084816681898677857653023} a^{4} - \frac{306298056129200116777635800939}{2171538508900091392067145918138} a^{3} - \frac{76082215583801395226667104932}{1085769254450045696033572959069} a^{2} + \frac{211625716610305450445634132608}{1085769254450045696033572959069} a - \frac{437596620246673253057436180728}{1085769254450045696033572959069}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 455641598745.8738 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), 4.0.2112.1, 5.1.4050000.3, 10.0.33592320000000000.77

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.10.19.33$x^{10} - 6 x^{4} + 4 x^{2} - 14$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
2.10.19.33$x^{10} - 6 x^{4} + 4 x^{2} - 14$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$