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Properties

Label 20.0.22089494171...4368.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 19^{10}$
Root discriminant $29.32$
Ramified primes $2, 19$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11576, 25888, 52056, 81312, 80510, 74528, 50816, 29992, 22413, 11660, 6254, 1036, 1741, 264, 220, -16, 63, -20, 14, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 14*x^18 - 20*x^17 + 63*x^16 - 16*x^15 + 220*x^14 + 264*x^13 + 1741*x^12 + 1036*x^11 + 6254*x^10 + 11660*x^9 + 22413*x^8 + 29992*x^7 + 50816*x^6 + 74528*x^5 + 80510*x^4 + 81312*x^3 + 52056*x^2 + 25888*x + 11576)
 
gp: K = bnfinit(x^20 - 4*x^19 + 14*x^18 - 20*x^17 + 63*x^16 - 16*x^15 + 220*x^14 + 264*x^13 + 1741*x^12 + 1036*x^11 + 6254*x^10 + 11660*x^9 + 22413*x^8 + 29992*x^7 + 50816*x^6 + 74528*x^5 + 80510*x^4 + 81312*x^3 + 52056*x^2 + 25888*x + 11576, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 14 x^{18} - 20 x^{17} + 63 x^{16} - 16 x^{15} + 220 x^{14} + 264 x^{13} + 1741 x^{12} + 1036 x^{11} + 6254 x^{10} + 11660 x^{9} + 22413 x^{8} + 29992 x^{7} + 50816 x^{6} + 74528 x^{5} + 80510 x^{4} + 81312 x^{3} + 52056 x^{2} + 25888 x + 11576 \)

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:  \(220894941712131239715817914368=2^{55}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.32$
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:  $2, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{12} - \frac{1}{16} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{32} a^{17} - \frac{1}{32} a^{16} - \frac{1}{16} a^{13} + \frac{1}{16} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{32} a^{9} - \frac{7}{32} a^{8} + \frac{3}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{16} a^{5} - \frac{3}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{512} a^{18} + \frac{1}{256} a^{17} - \frac{7}{512} a^{16} - \frac{13}{256} a^{14} + \frac{5}{128} a^{13} + \frac{23}{256} a^{12} + \frac{1}{64} a^{11} - \frac{113}{512} a^{10} - \frac{17}{256} a^{9} + \frac{51}{512} a^{8} - \frac{3}{16} a^{7} + \frac{125}{256} a^{6} + \frac{41}{128} a^{5} + \frac{79}{256} a^{4} + \frac{3}{32} a^{3} + \frac{3}{8} a^{2} - \frac{9}{32} a + \frac{7}{64}$, $\frac{1}{536993991139729532717295984901742107648} a^{19} + \frac{442477154075948006598368232629968259}{536993991139729532717295984901742107648} a^{18} + \frac{1125438254465417671007844123693633851}{536993991139729532717295984901742107648} a^{17} + \frac{1843511052448579530806173993953891071}{76713427305675647531042283557391729664} a^{16} - \frac{685539669477607737364479698424703531}{38356713652837823765521141778695864832} a^{15} + \frac{16256921798670527279871528273806927165}{268496995569864766358647992450871053824} a^{14} - \frac{27459120544920255698969490216655664159}{268496995569864766358647992450871053824} a^{13} + \frac{1060672963621161300303602267597329499}{268496995569864766358647992450871053824} a^{12} + \frac{50275889784483573233346816331349817751}{536993991139729532717295984901742107648} a^{11} + \frac{62686982180467133570177525541969334381}{536993991139729532717295984901742107648} a^{10} - \frac{34384794360293713330380143050497845167}{536993991139729532717295984901742107648} a^{9} - \frac{62501354496382484446527576049299814317}{536993991139729532717295984901742107648} a^{8} + \frac{8617041906481665075711381421382186413}{268496995569864766358647992450871053824} a^{7} + \frac{30571387162579904662097399362544841423}{268496995569864766358647992450871053824} a^{6} + \frac{39343097273003070403303236935152583713}{268496995569864766358647992450871053824} a^{5} - \frac{23920213420646743544480359284238572825}{268496995569864766358647992450871053824} a^{4} + \frac{15681727433369651778096388055665315863}{33562124446233095794830999056358881728} a^{3} + \frac{7094717749927338197124415887493541731}{33562124446233095794830999056358881728} a^{2} - \frac{17446959755949960159243341988159857003}{67124248892466191589661998112717763456} a + \frac{1233564548714001300800708320863063111}{67124248892466191589661998112717763456}$

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:  \( 20791866.5937 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.739328.2, 5.1.739328.1 x5, 10.2.4372847132672.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.739328.1
Degree 10 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.4.11.4$x^{4} + 12 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.4$x^{4} + 12 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.4$x^{4} + 12 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.4$x^{4} + 12 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.4$x^{4} + 12 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
$19$19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
19.4.2.2$x^{4} - 19 x^{2} + 722$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$