↑

Properties

Label 20.8.81693426134...7408.3
Degree $20$
Signature $[8, 6]$
Discriminant $2^{10}\cdot 3^{15}\cdot 11^{18}$
Root discriminant $27.90$
Ramified primes $2, 3, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T409

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![67, 547, 0, -5330, 579, 7546, -8412, 8945, -8657, 6615, -3642, 1884, -880, 149, -18, 22, -25, 20, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 20*x^17 - 25*x^16 + 22*x^15 - 18*x^14 + 149*x^13 - 880*x^12 + 1884*x^11 - 3642*x^10 + 6615*x^9 - 8657*x^8 + 8945*x^7 - 8412*x^6 + 7546*x^5 + 579*x^4 - 5330*x^3 + 547*x + 67)
 
gp: K = bnfinit(x^20 - 4*x^19 + 20*x^17 - 25*x^16 + 22*x^15 - 18*x^14 + 149*x^13 - 880*x^12 + 1884*x^11 - 3642*x^10 + 6615*x^9 - 8657*x^8 + 8945*x^7 - 8412*x^6 + 7546*x^5 + 579*x^4 - 5330*x^3 + 547*x + 67, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 20 x^{17} - 25 x^{16} + 22 x^{15} - 18 x^{14} + 149 x^{13} - 880 x^{12} + 1884 x^{11} - 3642 x^{10} + 6615 x^{9} - 8657 x^{8} + 8945 x^{7} - 8412 x^{6} + 7546 x^{5} + 579 x^{4} - 5330 x^{3} + 547 x + 67 \)

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:  \(81693426134005631737181457408=2^{10}\cdot 3^{15}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.90$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{67} a^{18} - \frac{13}{67} a^{17} + \frac{13}{67} a^{16} - \frac{18}{67} a^{15} - \frac{9}{67} a^{14} + \frac{32}{67} a^{13} + \frac{27}{67} a^{12} - \frac{5}{67} a^{11} - \frac{25}{67} a^{10} + \frac{16}{67} a^{9} + \frac{20}{67} a^{8} + \frac{14}{67} a^{7} - \frac{9}{67} a^{6} - \frac{1}{67} a^{5} - \frac{30}{67} a^{4} + \frac{14}{67} a^{3} + \frac{22}{67} a^{2} - \frac{16}{67} a$, $\frac{1}{711835037648307603749682659449626903097} a^{19} - \frac{3565824299736789856621884936271373344}{711835037648307603749682659449626903097} a^{18} + \frac{230151707656562478479184417648946745080}{711835037648307603749682659449626903097} a^{17} + \frac{225516632546430573943568504763189656362}{711835037648307603749682659449626903097} a^{16} - \frac{331268099161178082341377833759688868956}{711835037648307603749682659449626903097} a^{15} - \frac{33431231197333477271368427264108951712}{711835037648307603749682659449626903097} a^{14} - \frac{130352773152102799057934746907609705777}{711835037648307603749682659449626903097} a^{13} - \frac{77344758933889258585043079045520517582}{711835037648307603749682659449626903097} a^{12} + \frac{186890869843537289445426567706520430631}{711835037648307603749682659449626903097} a^{11} - \frac{250181548664640860683329899444051366678}{711835037648307603749682659449626903097} a^{10} - \frac{41981780320425420532830033956018181489}{711835037648307603749682659449626903097} a^{9} + \frac{287068571912694296686848879731034370726}{711835037648307603749682659449626903097} a^{8} + \frac{260793774792757199006519618890985499961}{711835037648307603749682659449626903097} a^{7} - \frac{52319903906732184628906745971560919501}{711835037648307603749682659449626903097} a^{6} + \frac{206275538894269200266785071133477000442}{711835037648307603749682659449626903097} a^{5} + \frac{224584836224568277225398265599333180750}{711835037648307603749682659449626903097} a^{4} - \frac{119955257789605007805098797534149247609}{711835037648307603749682659449626903097} a^{3} + \frac{113700547843843244807700369770467392036}{711835037648307603749682659449626903097} a^{2} + \frac{116604387019355997036370068835143747455}{711835037648307603749682659449626903097} a - \frac{2653576064378555693533312511782225946}{10624403546989665727607203872382491091}$

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:  \( 5659848.98043 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T409:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n409 are not computed
Character table for t20n409 is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{33})^+\)

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 R $20$ $20$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.10.9$x^{10} - 15 x^{8} + 38 x^{6} - 18 x^{4} + 25 x^{2} - 63$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.0.1$x^{10} - x^{3} + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
3Data not computed
11Data not computed