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Properties

Label 20.20.7303816416...9296.6
Degree $20$
Signature $[20, 0]$
Discriminant $2^{30}\cdot 11^{16}\cdot 23^{6}$
Root discriminant $49.34$
Ramified primes $2, 11, 23$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^4:C_5$ (as 20T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16928, -169280, 712224, -1585536, 1818032, -476416, -1394144, 1665312, -326024, -649264, 451848, 20384, -120232, 30864, 10688, -5752, 74, 340, -40, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 - 40*x^18 + 340*x^17 + 74*x^16 - 5752*x^15 + 10688*x^14 + 30864*x^13 - 120232*x^12 + 20384*x^11 + 451848*x^10 - 649264*x^9 - 326024*x^8 + 1665312*x^7 - 1394144*x^6 - 476416*x^5 + 1818032*x^4 - 1585536*x^3 + 712224*x^2 - 169280*x + 16928)
 
gp: K = bnfinit(x^20 - 6*x^19 - 40*x^18 + 340*x^17 + 74*x^16 - 5752*x^15 + 10688*x^14 + 30864*x^13 - 120232*x^12 + 20384*x^11 + 451848*x^10 - 649264*x^9 - 326024*x^8 + 1665312*x^7 - 1394144*x^6 - 476416*x^5 + 1818032*x^4 - 1585536*x^3 + 712224*x^2 - 169280*x + 16928, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} - 40 x^{18} + 340 x^{17} + 74 x^{16} - 5752 x^{15} + 10688 x^{14} + 30864 x^{13} - 120232 x^{12} + 20384 x^{11} + 451848 x^{10} - 649264 x^{9} - 326024 x^{8} + 1665312 x^{7} - 1394144 x^{6} - 476416 x^{5} + 1818032 x^{4} - 1585536 x^{3} + 712224 x^{2} - 169280 x + 16928 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7303816416639774784171798530359296=2^{30}\cdot 11^{16}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.34$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{88} a^{15} + \frac{1}{22} a^{14} + \frac{1}{22} a^{13} + \frac{3}{88} a^{12} + \frac{1}{11} a^{11} + \frac{1}{22} a^{10} + \frac{3}{44} a^{9} + \frac{1}{11} a^{8} - \frac{3}{22} a^{7} - \frac{3}{22} a^{6} - \frac{1}{22} a^{4} - \frac{4}{11} a^{3} - \frac{3}{11} a^{2} + \frac{3}{11} a + \frac{4}{11}$, $\frac{1}{176} a^{16} + \frac{5}{88} a^{14} - \frac{1}{88} a^{13} - \frac{1}{44} a^{12} + \frac{1}{11} a^{11} + \frac{3}{44} a^{10} - \frac{1}{11} a^{9} - \frac{1}{22} a^{7} - \frac{5}{22} a^{6} + \frac{5}{22} a^{5} - \frac{1}{11} a^{4} + \frac{1}{11} a^{3} + \frac{2}{11} a^{2} - \frac{4}{11} a + \frac{3}{11}$, $\frac{1}{4048} a^{17} + \frac{5}{4048} a^{16} + \frac{1}{1012} a^{15} - \frac{27}{1012} a^{14} - \frac{12}{253} a^{13} + \frac{3}{92} a^{12} + \frac{1}{23} a^{11} - \frac{3}{506} a^{10} - \frac{9}{506} a^{9} + \frac{19}{1012} a^{8} - \frac{45}{506} a^{7} + \frac{4}{23} a^{6} - \frac{49}{253} a^{5} + \frac{61}{506} a^{4} - \frac{113}{253} a^{3} - \frac{40}{253} a^{2} + \frac{51}{253} a + \frac{3}{11}$, $\frac{1}{4048} a^{18} + \frac{1}{2024} a^{16} + \frac{5}{2024} a^{15} + \frac{59}{2024} a^{14} + \frac{5}{253} a^{13} - \frac{81}{2024} a^{12} - \frac{111}{1012} a^{11} - \frac{17}{506} a^{10} - \frac{29}{1012} a^{9} + \frac{91}{1012} a^{8} + \frac{83}{506} a^{7} - \frac{101}{506} a^{6} - \frac{93}{506} a^{5} + \frac{113}{506} a^{4} + \frac{19}{253} a^{3} + \frac{90}{253} a^{2} - \frac{71}{253} a$, $\frac{1}{4048} a^{19} + \frac{9}{2024} a^{15} - \frac{9}{506} a^{14} - \frac{73}{2024} a^{13} + \frac{7}{1012} a^{12} - \frac{53}{1012} a^{11} - \frac{109}{1012} a^{10} - \frac{1}{92} a^{9} - \frac{14}{253} a^{8} - \frac{63}{253} a^{7} + \frac{61}{253} a^{6} + \frac{28}{253} a^{5} - \frac{19}{253} a^{4} - \frac{6}{253} a^{3} - \frac{106}{253} a^{2} + \frac{13}{253} a - \frac{3}{11}$

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:  $19$
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:  \( 43517716745.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^4:C_5$ (as 20T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 16 conjugacy class representatives for $C_2\times C_2^4:C_5$
Character table for $C_2\times C_2^4:C_5$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.5048580365312.1, 10.10.2670699013250048.1, 10.10.116117348402176.2

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 32 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$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}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$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}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$