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Properties

Label 20.12.1948135206...8125.2
Degree $20$
Signature $[12, 4]$
Discriminant $3^{8}\cdot 5^{11}\cdot 239^{10}$
Root discriminant $58.14$
Ramified primes $3, 5, 239$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-759959, 2984623, -3430589, -913289, 4733767, -2635848, -1448287, 2178107, -419884, -626572, 358342, 49241, -87949, 11555, 9767, -2967, -395, 256, -11, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 11*x^18 + 256*x^17 - 395*x^16 - 2967*x^15 + 9767*x^14 + 11555*x^13 - 87949*x^12 + 49241*x^11 + 358342*x^10 - 626572*x^9 - 419884*x^8 + 2178107*x^7 - 1448287*x^6 - 2635848*x^5 + 4733767*x^4 - 913289*x^3 - 3430589*x^2 + 2984623*x - 759959)
 
gp: K = bnfinit(x^20 - 8*x^19 - 11*x^18 + 256*x^17 - 395*x^16 - 2967*x^15 + 9767*x^14 + 11555*x^13 - 87949*x^12 + 49241*x^11 + 358342*x^10 - 626572*x^9 - 419884*x^8 + 2178107*x^7 - 1448287*x^6 - 2635848*x^5 + 4733767*x^4 - 913289*x^3 - 3430589*x^2 + 2984623*x - 759959, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 11 x^{18} + 256 x^{17} - 395 x^{16} - 2967 x^{15} + 9767 x^{14} + 11555 x^{13} - 87949 x^{12} + 49241 x^{11} + 358342 x^{10} - 626572 x^{9} - 419884 x^{8} + 2178107 x^{7} - 1448287 x^{6} - 2635848 x^{5} + 4733767 x^{4} - 913289 x^{3} - 3430589 x^{2} + 2984623 x - 759959 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(194813520645626483248608015673828125=3^{8}\cdot 5^{11}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.14$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{15} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{344743395658878834502151832822051479189430591297} a^{19} - \frac{35138131100866236136630109403585511741873738748}{344743395658878834502151832822051479189430591297} a^{18} - \frac{51971579543178929560631742535418045870692122511}{344743395658878834502151832822051479189430591297} a^{17} - \frac{27487624070627811541740271276528901439563503561}{344743395658878834502151832822051479189430591297} a^{16} - \frac{6835873858265962478943373237373788706788395011}{114914465219626278167383944274017159729810197099} a^{15} - \frac{40102811224870845075601975271205371813882222560}{344743395658878834502151832822051479189430591297} a^{14} - \frac{139449941101643292505345215209895924147876572611}{344743395658878834502151832822051479189430591297} a^{13} - \frac{20883460048130494698201840050590656140522182775}{344743395658878834502151832822051479189430591297} a^{12} - \frac{56813295165340348907023268986811218294777351277}{344743395658878834502151832822051479189430591297} a^{11} - \frac{53832611370269929388048379359144740033570208279}{114914465219626278167383944274017159729810197099} a^{10} + \frac{96961097989895320380932535425368551794440590155}{344743395658878834502151832822051479189430591297} a^{9} - \frac{241573101743257758735140468536255683357620819}{1925940757870831477665652697329896531784528443} a^{8} + \frac{165569566821649394058627304173750897312189784186}{344743395658878834502151832822051479189430591297} a^{7} + \frac{24347385094663160989043576205145383020883407749}{344743395658878834502151832822051479189430591297} a^{6} - \frac{56355169899373003064802667719318698137378329200}{114914465219626278167383944274017159729810197099} a^{5} + \frac{74625062047549596849542897349263523096047522968}{344743395658878834502151832822051479189430591297} a^{4} - \frac{107041619603536761757943932703534890898858990954}{344743395658878834502151832822051479189430591297} a^{3} - \frac{155353706126340220987332895875777349373159728843}{344743395658878834502151832822051479189430591297} a^{2} - \frac{121386639040230810656441238955867103704616640469}{344743395658878834502151832822051479189430591297} a + \frac{28728692380125617680010986891744947211357438882}{114914465219626278167383944274017159729810197099}$

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:  $15$
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:  \( 18449348920.5 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 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 $20$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed