↑

Properties

Label 20.8.49916992757...0912.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{54}\cdot 31^{4}\cdot 113\cdot 227^{4}$
Root discriminant $48.41$
Ramified primes $2, 31, 113, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -56, -200, 88, 786, 824, 116, -248, -193, -388, -516, -400, -221, -116, -88, -16, -1, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 4*x^18 - x^16 - 16*x^15 - 88*x^14 - 116*x^13 - 221*x^12 - 400*x^11 - 516*x^10 - 388*x^9 - 193*x^8 - 248*x^7 + 116*x^6 + 824*x^5 + 786*x^4 + 88*x^3 - 200*x^2 - 56*x + 2)
 
gp: K = bnfinit(x^20 + 4*x^18 - x^16 - 16*x^15 - 88*x^14 - 116*x^13 - 221*x^12 - 400*x^11 - 516*x^10 - 388*x^9 - 193*x^8 - 248*x^7 + 116*x^6 + 824*x^5 + 786*x^4 + 88*x^3 - 200*x^2 - 56*x + 2, 1)
 

Normalized defining polynomial

\( x^{20} + 4 x^{18} - x^{16} - 16 x^{15} - 88 x^{14} - 116 x^{13} - 221 x^{12} - 400 x^{11} - 516 x^{10} - 388 x^{9} - 193 x^{8} - 248 x^{7} + 116 x^{6} + 824 x^{5} + 786 x^{4} + 88 x^{3} - 200 x^{2} - 56 x + 2 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4991699275727550043990996300070912=2^{54}\cdot 31^{4}\cdot 113\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.41$
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:  $2, 31, 113, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{93085976828187792679211} a^{19} + \frac{13181137466112085093679}{93085976828187792679211} a^{18} - \frac{27894635501342115428379}{93085976828187792679211} a^{17} + \frac{24557737747324443540464}{93085976828187792679211} a^{16} - \frac{20784743398658390483761}{93085976828187792679211} a^{15} + \frac{42653350318460706550655}{93085976828187792679211} a^{14} - \frac{9356516959046633526436}{93085976828187792679211} a^{13} - \frac{43814484060629708286388}{93085976828187792679211} a^{12} + \frac{265715453480216733556}{538069230220738686007} a^{11} - \frac{42264032006616882043273}{93085976828187792679211} a^{10} - \frac{14917505779902644412483}{93085976828187792679211} a^{9} + \frac{31492473113741298830454}{93085976828187792679211} a^{8} + \frac{23428598026144002580188}{93085976828187792679211} a^{7} + \frac{12893017391828921988130}{93085976828187792679211} a^{6} + \frac{42023907951798501706669}{93085976828187792679211} a^{5} + \frac{30442230692301622487006}{93085976828187792679211} a^{4} + \frac{28963452136492804415356}{93085976828187792679211} a^{3} + \frac{34933642518944028673418}{93085976828187792679211} a^{2} - \frac{14182680192323073561863}{93085976828187792679211} a - \frac{9898139076542029856454}{93085976828187792679211}$

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:  $13$
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:  \( 5353596238.48 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $16{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.8.26.1$x^{8} + 4 x^{6} + 8 x^{3} + 8 x^{2} + 2$$8$$1$$26$$C_2^2:C_4$$[2, 3, 7/2, 4]$
2.12.28.57$x^{12} + 2 x^{10} - 2 x^{8} + 4 x^{5} - 2 x^{4} - 2$$12$$1$$28$12T48$[2, 8/3, 8/3, 3]_{3}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$
31.5.0.1$x^{5} - x + 10$$1$$5$$0$$C_5$$[\ ]^{5}$
31.6.4.1$x^{6} + 1085 x^{3} + 1660608$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$113$113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.1.2$x^{2} + 339$$2$$1$$1$$C_2$$[\ ]_{2}$
113.4.0.1$x^{4} - x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
113.5.0.1$x^{5} - x + 17$$1$$5$$0$$C_5$$[\ ]^{5}$
113.5.0.1$x^{5} - x + 17$$1$$5$$0$$C_5$$[\ ]^{5}$
227Data not computed