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Properties

Label 20.6.14672147998...9375.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,3^{10}\cdot 5^{10}\cdot 239^{9}$
Root discriminant $45.53$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3249, -1938, 34750, -60453, 32384, 17141, -35775, 20087, -196, -7092, 7741, -2961, -1972, 2196, -341, -626, 169, 81, -22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 22*x^18 + 81*x^17 + 169*x^16 - 626*x^15 - 341*x^14 + 2196*x^13 - 1972*x^12 - 2961*x^11 + 7741*x^10 - 7092*x^9 - 196*x^8 + 20087*x^7 - 35775*x^6 + 17141*x^5 + 32384*x^4 - 60453*x^3 + 34750*x^2 - 1938*x - 3249)
 
gp: K = bnfinit(x^20 - 4*x^19 - 22*x^18 + 81*x^17 + 169*x^16 - 626*x^15 - 341*x^14 + 2196*x^13 - 1972*x^12 - 2961*x^11 + 7741*x^10 - 7092*x^9 - 196*x^8 + 20087*x^7 - 35775*x^6 + 17141*x^5 + 32384*x^4 - 60453*x^3 + 34750*x^2 - 1938*x - 3249, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 22 x^{18} + 81 x^{17} + 169 x^{16} - 626 x^{15} - 341 x^{14} + 2196 x^{13} - 1972 x^{12} - 2961 x^{11} + 7741 x^{10} - 7092 x^{9} - 196 x^{8} + 20087 x^{7} - 35775 x^{6} + 17141 x^{5} + 32384 x^{4} - 60453 x^{3} + 34750 x^{2} - 1938 x - 3249 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1467214799841538367562738193359375=-\,3^{10}\cdot 5^{10}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.53$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{12} - \frac{3}{10} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{10} a^{6} + \frac{1}{10} a^{5} + \frac{3}{10} a^{4} - \frac{3}{10} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{3}{10}$, $\frac{1}{10} a^{13} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{3}{10} a^{7} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{5} a^{2} + \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{13} - \frac{1}{20} a^{12} + \frac{1}{20} a^{11} + \frac{1}{20} a^{10} + \frac{9}{20} a^{9} - \frac{1}{4} a^{8} + \frac{2}{5} a^{7} - \frac{7}{20} a^{6} + \frac{3}{20} a^{5} - \frac{1}{10} a^{4} + \frac{7}{20} a^{3} - \frac{3}{20} a^{2} + \frac{3}{20} a - \frac{7}{20}$, $\frac{1}{60} a^{15} + \frac{1}{60} a^{14} + \frac{1}{60} a^{13} + \frac{1}{60} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} + \frac{19}{60} a^{9} + \frac{3}{10} a^{8} + \frac{3}{20} a^{7} - \frac{17}{60} a^{6} + \frac{13}{30} a^{5} - \frac{11}{60} a^{4} - \frac{9}{20} a^{3} + \frac{1}{12} a^{2} - \frac{29}{60} a + \frac{1}{5}$, $\frac{1}{120} a^{16} - \frac{1}{40} a^{14} - \frac{1}{40} a^{13} + \frac{1}{40} a^{12} + \frac{3}{40} a^{11} + \frac{1}{40} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{4}{15} a^{7} + \frac{2}{15} a^{6} + \frac{19}{60} a^{5} + \frac{5}{12} a^{4} - \frac{31}{120} a^{3} + \frac{59}{120} a^{2} - \frac{1}{12} a + \frac{13}{40}$, $\frac{1}{120} a^{17} - \frac{1}{120} a^{15} - \frac{1}{120} a^{14} + \frac{1}{24} a^{13} - \frac{1}{120} a^{12} - \frac{7}{120} a^{11} + \frac{1}{30} a^{10} - \frac{1}{30} a^{9} + \frac{7}{30} a^{8} - \frac{7}{60} a^{7} - \frac{1}{15} a^{6} - \frac{1}{20} a^{5} + \frac{7}{120} a^{4} - \frac{11}{24} a^{3} - \frac{1}{5} a^{2} + \frac{1}{24} a - \frac{3}{10}$, $\frac{1}{4560} a^{18} - \frac{7}{4560} a^{17} + \frac{3}{760} a^{16} - \frac{1}{152} a^{15} + \frac{23}{1520} a^{14} - \frac{37}{1520} a^{13} + \frac{29}{1520} a^{12} - \frac{11}{2280} a^{11} - \frac{101}{4560} a^{10} + \frac{409}{1140} a^{9} - \frac{367}{760} a^{8} + \frac{23}{2280} a^{7} + \frac{83}{380} a^{6} + \frac{519}{1520} a^{5} - \frac{91}{760} a^{4} + \frac{283}{760} a^{3} + \frac{53}{570} a^{2} - \frac{549}{1520} a - \frac{1}{80}$, $\frac{1}{33072127179928346334960} a^{19} + \frac{2556046326470474617}{33072127179928346334960} a^{18} - \frac{257255848593979769}{123403459626598307220} a^{17} - \frac{9714981541299577421}{4134015897491043291870} a^{16} + \frac{54291000471161850371}{33072127179928346334960} a^{15} - \frac{768599475228039324859}{33072127179928346334960} a^{14} - \frac{539056436688009499033}{33072127179928346334960} a^{13} + \frac{727102269843223878581}{16536063589964173167480} a^{12} - \frac{1761977466635749703861}{33072127179928346334960} a^{11} + \frac{838713839935651333051}{16536063589964173167480} a^{10} - \frac{647515283681956050119}{16536063589964173167480} a^{9} + \frac{8209152974844321756869}{16536063589964173167480} a^{8} - \frac{1512116308063702852561}{4134015897491043291870} a^{7} + \frac{1390478695864587879541}{33072127179928346334960} a^{6} + \frac{3089680725698317855963}{16536063589964173167480} a^{5} + \frac{581086214803892741167}{2756010598327362194580} a^{4} - \frac{665998585620824323678}{2067007948745521645935} a^{3} - \frac{7846727475778579093109}{33072127179928346334960} a^{2} + \frac{2669598356723028002927}{11024042393309448778320} a + \frac{25402289831078015851}{96702126257100427880}$

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:  $12$
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:  \( 626711065.375 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed