↑

Properties

Label 20.0.19544604289...5129.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{6}\cdot 401^{9}$
Root discriminant $20.63$
Ramified primes $3, 401$
Class number $2$
Class group $[2]$
Galois group $C_2^4:D_5$ (as 20T38)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 16, -24, 23, -7, 21, -75, 188, -212, 303, -320, 214, -64, -5, 14, 15, -11, 10, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 10*x^18 - 11*x^17 + 15*x^16 + 14*x^15 - 5*x^14 - 64*x^13 + 214*x^12 - 320*x^11 + 303*x^10 - 212*x^9 + 188*x^8 - 75*x^7 + 21*x^6 - 7*x^5 + 23*x^4 - 24*x^3 + 16*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^20 - 2*x^19 + 10*x^18 - 11*x^17 + 15*x^16 + 14*x^15 - 5*x^14 - 64*x^13 + 214*x^12 - 320*x^11 + 303*x^10 - 212*x^9 + 188*x^8 - 75*x^7 + 21*x^6 - 7*x^5 + 23*x^4 - 24*x^3 + 16*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 10 x^{18} - 11 x^{17} + 15 x^{16} + 14 x^{15} - 5 x^{14} - 64 x^{13} + 214 x^{12} - 320 x^{11} + 303 x^{10} - 212 x^{9} + 188 x^{8} - 75 x^{7} + 21 x^{6} - 7 x^{5} + 23 x^{4} - 24 x^{3} + 16 x^{2} - 5 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(195446042895196345705665129=3^{6}\cdot 401^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.63$
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:  $3, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{17} + \frac{1}{6} a^{16} + \frac{1}{6} a^{15} + \frac{1}{3} a^{14} + \frac{1}{6} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{6} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{18} a^{18} - \frac{1}{9} a^{15} - \frac{7}{18} a^{14} - \frac{7}{18} a^{13} - \frac{1}{6} a^{12} - \frac{1}{2} a^{11} + \frac{1}{18} a^{10} + \frac{1}{3} a^{9} - \frac{1}{18} a^{8} - \frac{2}{9} a^{7} + \frac{1}{18} a^{6} + \frac{1}{6} a^{5} - \frac{1}{18} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{67423215994305642} a^{19} + \frac{310932985884070}{33711607997152821} a^{18} + \frac{296622200589091}{3745734221905869} a^{17} + \frac{3586443777306835}{67423215994305642} a^{16} - \frac{6131281601234914}{33711607997152821} a^{15} + \frac{1134192417273943}{2497156147937246} a^{14} + \frac{13353400763420683}{67423215994305642} a^{13} + \frac{323871412340864}{11237202665717607} a^{12} + \frac{26601310078515613}{67423215994305642} a^{11} - \frac{15454810713379961}{33711607997152821} a^{10} - \frac{6180432060340033}{67423215994305642} a^{9} - \frac{635663390569445}{11237202665717607} a^{8} + \frac{4204128725918980}{33711607997152821} a^{7} - \frac{16086310049853233}{33711607997152821} a^{6} - \frac{14455222427101133}{33711607997152821} a^{5} - \frac{15657790915010827}{33711607997152821} a^{4} + \frac{3389689935642398}{11237202665717607} a^{3} + \frac{1936953252856027}{11237202665717607} a^{2} - \frac{5750253383954182}{33711607997152821} a - \frac{42560549563357}{67423215994305642}$

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:  \( 33411.4738636 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:D_5$ (as 20T38):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 10 conjugacy class representatives for $C_2^4:D_5$
Character table for $C_2^4:D_5$

Intermediate fields

5.5.160801.1, 10.6.232712654409.1

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 16 sibling: 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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
401Data not computed