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Properties

Label 20.0.49668756201...6832.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 13^{10}$
Root discriminant $24.26$
Ramified primes $2, 13$
Class number $2$
Class group $[2]$
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![16, -96, 168, -72, 1047, -1304, 1924, -2744, 4326, -4352, 2600, -32, -1232, 1128, -380, -88, 194, -112, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 40*x^18 - 112*x^17 + 194*x^16 - 88*x^15 - 380*x^14 + 1128*x^13 - 1232*x^12 - 32*x^11 + 2600*x^10 - 4352*x^9 + 4326*x^8 - 2744*x^7 + 1924*x^6 - 1304*x^5 + 1047*x^4 - 72*x^3 + 168*x^2 - 96*x + 16)
 
gp: K = bnfinit(x^20 - 8*x^19 + 40*x^18 - 112*x^17 + 194*x^16 - 88*x^15 - 380*x^14 + 1128*x^13 - 1232*x^12 - 32*x^11 + 2600*x^10 - 4352*x^9 + 4326*x^8 - 2744*x^7 + 1924*x^6 - 1304*x^5 + 1047*x^4 - 72*x^3 + 168*x^2 - 96*x + 16, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 40 x^{18} - 112 x^{17} + 194 x^{16} - 88 x^{15} - 380 x^{14} + 1128 x^{13} - 1232 x^{12} - 32 x^{11} + 2600 x^{10} - 4352 x^{9} + 4326 x^{8} - 2744 x^{7} + 1924 x^{6} - 1304 x^{5} + 1047 x^{4} - 72 x^{3} + 168 x^{2} - 96 x + 16 \)

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:  \(4966875620168119680952696832=2^{55}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.26$
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:  $2, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{11} + \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{12} + \frac{3}{8} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{16} a^{5} + \frac{3}{16} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{14} + \frac{1}{32} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{32} a^{6} - \frac{1}{8} a^{5} + \frac{11}{32} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{128} a^{15} - \frac{1}{128} a^{14} - \frac{3}{128} a^{13} - \frac{1}{128} a^{12} + \frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} - \frac{5}{128} a^{7} - \frac{19}{128} a^{6} - \frac{29}{128} a^{5} - \frac{55}{128} a^{4} + \frac{1}{4} a^{3} - \frac{7}{16} a^{2} - \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{128} a^{16} - \frac{1}{32} a^{13} + \frac{7}{128} a^{12} + \frac{3}{32} a^{10} - \frac{1}{16} a^{9} + \frac{3}{128} a^{8} + \frac{3}{16} a^{7} - \frac{5}{32} a^{6} - \frac{1}{32} a^{5} + \frac{53}{128} a^{4} + \frac{3}{16} a^{3} - \frac{5}{16} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{256} a^{17} - \frac{1}{256} a^{16} - \frac{1}{64} a^{14} - \frac{5}{256} a^{13} - \frac{7}{256} a^{12} - \frac{1}{64} a^{11} + \frac{3}{64} a^{10} + \frac{11}{256} a^{9} + \frac{21}{256} a^{8} - \frac{11}{64} a^{7} - \frac{3}{16} a^{6} - \frac{55}{256} a^{5} + \frac{99}{256} a^{4} + \frac{5}{16} a^{3} + \frac{13}{32} a^{2} - \frac{7}{16} a - \frac{3}{16}$, $\frac{1}{209681408} a^{18} - \frac{72097}{104840704} a^{17} + \frac{87731}{209681408} a^{16} - \frac{26183}{52420352} a^{15} - \frac{1688929}{209681408} a^{14} - \frac{456209}{104840704} a^{13} + \frac{2583297}{209681408} a^{12} + \frac{312333}{13105088} a^{11} - \frac{241705}{2161664} a^{10} - \frac{12175827}{104840704} a^{9} + \frac{574133}{209681408} a^{8} - \frac{2808767}{52420352} a^{7} + \frac{1783313}{209681408} a^{6} - \frac{19934171}{104840704} a^{5} + \frac{85844359}{209681408} a^{4} - \frac{2773807}{26210176} a^{3} - \frac{8257701}{26210176} a^{2} + \frac{14739}{204767} a - \frac{4888791}{13105088}$, $\frac{1}{838725632} a^{19} - \frac{1}{838725632} a^{18} - \frac{455599}{838725632} a^{17} + \frac{405159}{838725632} a^{16} + \frac{303715}{838725632} a^{15} - \frac{7353955}{838725632} a^{14} - \frac{267393}{8646656} a^{13} - \frac{3955935}{838725632} a^{12} + \frac{38532679}{838725632} a^{11} + \frac{100325841}{838725632} a^{10} + \frac{7044911}{838725632} a^{9} - \frac{46155287}{838725632} a^{8} - \frac{155422027}{838725632} a^{7} + \frac{143370107}{838725632} a^{6} + \frac{14803345}{838725632} a^{5} + \frac{79710079}{838725632} a^{4} + \frac{9516711}{26210176} a^{3} - \frac{15089557}{104840704} a^{2} - \frac{25574647}{52420352} a + \frac{16176761}{52420352}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3641112.82209 \)
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

\(\Q(\sqrt{2}) \), 4.0.346112.2, 5.1.346112.1 x5, 10.2.958348132352.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.346112.1
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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.3$x^{4} + 4 x^{2} + 18$$4$$1$$11$$C_4$$[3, 4]$
$13$13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$