↑

Properties

Label 20.8.33313079744...9872.3
Degree $20$
Signature $[8, 6]$
Discriminant $2^{10}\cdot 11^{17}\cdot 23^{5}$
Root discriminant $23.78$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T749

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -40, -183, 386, 461, -1113, -284, 1703, -478, -1034, 232, 902, -378, -473, 388, 23, -119, 36, 7, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 7*x^18 + 36*x^17 - 119*x^16 + 23*x^15 + 388*x^14 - 473*x^13 - 378*x^12 + 902*x^11 + 232*x^10 - 1034*x^9 - 478*x^8 + 1703*x^7 - 284*x^6 - 1113*x^5 + 461*x^4 + 386*x^3 - 183*x^2 - 40*x + 1)
 
gp: K = bnfinit(x^20 - 6*x^19 + 7*x^18 + 36*x^17 - 119*x^16 + 23*x^15 + 388*x^14 - 473*x^13 - 378*x^12 + 902*x^11 + 232*x^10 - 1034*x^9 - 478*x^8 + 1703*x^7 - 284*x^6 - 1113*x^5 + 461*x^4 + 386*x^3 - 183*x^2 - 40*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 7 x^{18} + 36 x^{17} - 119 x^{16} + 23 x^{15} + 388 x^{14} - 473 x^{13} - 378 x^{12} + 902 x^{11} + 232 x^{10} - 1034 x^{9} - 478 x^{8} + 1703 x^{7} - 284 x^{6} - 1113 x^{5} + 461 x^{4} + 386 x^{3} - 183 x^{2} - 40 x + 1 \)

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:  \(3331307974402283487149519872=2^{10}\cdot 11^{17}\cdot 23^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.78$
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:  $2, 11, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} - \frac{1}{4} a^{12} + \frac{1}{8} a^{11} - \frac{3}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{2} a^{8} - \frac{3}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{15} - \frac{1}{4} a^{13} + \frac{1}{8} a^{12} - \frac{3}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{2} a^{9} - \frac{3}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} + \frac{3}{8} a$, $\frac{1}{16} a^{18} + \frac{1}{16} a^{14} + \frac{1}{16} a^{13} + \frac{3}{16} a^{12} - \frac{3}{8} a^{11} - \frac{7}{16} a^{10} - \frac{3}{8} a^{9} - \frac{7}{16} a^{8} + \frac{3}{8} a^{7} - \frac{5}{16} a^{6} + \frac{3}{16} a^{5} + \frac{5}{16} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{3}{8} a + \frac{7}{16}$, $\frac{1}{35641682089766939632} a^{19} + \frac{224098317572681069}{8910420522441734908} a^{18} - \frac{517420322653860881}{8910420522441734908} a^{17} - \frac{85178460406032654}{2227605130610433727} a^{16} - \frac{2057341376465590855}{35641682089766939632} a^{15} + \frac{1983436513785599297}{35641682089766939632} a^{14} - \frac{5385931232112667161}{35641682089766939632} a^{13} + \frac{2254505657863919389}{17820841044883469816} a^{12} + \frac{15574803580674597277}{35641682089766939632} a^{11} - \frac{8396176553098426337}{17820841044883469816} a^{10} - \frac{129932943828541253}{531965404324879696} a^{9} + \frac{4533349916809911737}{17820841044883469816} a^{8} + \frac{5290123453225741067}{35641682089766939632} a^{7} - \frac{11038179879335223693}{35641682089766939632} a^{6} - \frac{10319753573042136435}{35641682089766939632} a^{5} + \frac{1046775056440991075}{8910420522441734908} a^{4} - \frac{2228582905814117045}{17820841044883469816} a^{3} + \frac{2411413716283965457}{17820841044883469816} a^{2} + \frac{9932795776184218631}{35641682089766939632} a + \frac{734523616666761861}{2227605130610433727}$

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:  \( 1403074.79518 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T749:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 81920
The 332 conjugacy class representatives for t20n749 are not computed
Character table for t20n749 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.113395848049.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.6$x^{10} - 5 x^{8} - 18 x^{6} - 46 x^{4} + 49 x^{2} - 13$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2]^{10}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.9.7$x^{10} + 2673$$10$$1$$9$$C_{10}$$[\ ]_{10}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$