↑

Properties

Label 20.20.7303816416...9296.5
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)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-263, -43810, -2608, 296450, 156250, -639678, -505908, 561074, 584299, -208840, -319628, 20666, 91351, 5956, -14226, -1658, 1215, 142, -54, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 54*x^18 + 142*x^17 + 1215*x^16 - 1658*x^15 - 14226*x^14 + 5956*x^13 + 91351*x^12 + 20666*x^11 - 319628*x^10 - 208840*x^9 + 584299*x^8 + 561074*x^7 - 505908*x^6 - 639678*x^5 + 156250*x^4 + 296450*x^3 - 2608*x^2 - 43810*x - 263)
 
gp: K = bnfinit(x^20 - 4*x^19 - 54*x^18 + 142*x^17 + 1215*x^16 - 1658*x^15 - 14226*x^14 + 5956*x^13 + 91351*x^12 + 20666*x^11 - 319628*x^10 - 208840*x^9 + 584299*x^8 + 561074*x^7 - 505908*x^6 - 639678*x^5 + 156250*x^4 + 296450*x^3 - 2608*x^2 - 43810*x - 263, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 54 x^{18} + 142 x^{17} + 1215 x^{16} - 1658 x^{15} - 14226 x^{14} + 5956 x^{13} + 91351 x^{12} + 20666 x^{11} - 319628 x^{10} - 208840 x^{9} + 584299 x^{8} + 561074 x^{7} - 505908 x^{6} - 639678 x^{5} + 156250 x^{4} + 296450 x^{3} - 2608 x^{2} - 43810 x - 263 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{77} a^{15} + \frac{25}{77} a^{14} + \frac{9}{77} a^{13} - \frac{16}{77} a^{12} + \frac{13}{77} a^{11} + \frac{1}{7} a^{10} + \frac{16}{77} a^{9} - \frac{24}{77} a^{8} + \frac{36}{77} a^{7} - \frac{32}{77} a^{6} + \frac{3}{7} a^{5} - \frac{12}{77} a^{4} - \frac{26}{77} a^{3} - \frac{19}{77} a^{2} - \frac{6}{77} a + \frac{1}{77}$, $\frac{1}{77} a^{16} - \frac{10}{77} a^{13} + \frac{4}{11} a^{12} - \frac{6}{77} a^{11} - \frac{4}{11} a^{10} + \frac{38}{77} a^{9} + \frac{20}{77} a^{8} - \frac{8}{77} a^{7} - \frac{2}{11} a^{6} + \frac{10}{77} a^{5} - \frac{34}{77} a^{4} + \frac{15}{77} a^{3} + \frac{1}{11} a^{2} - \frac{3}{77} a - \frac{25}{77}$, $\frac{1}{77} a^{17} - \frac{10}{77} a^{14} + \frac{4}{11} a^{13} - \frac{6}{77} a^{12} - \frac{4}{11} a^{11} + \frac{38}{77} a^{10} + \frac{20}{77} a^{9} - \frac{8}{77} a^{8} - \frac{2}{11} a^{7} + \frac{10}{77} a^{6} - \frac{34}{77} a^{5} + \frac{15}{77} a^{4} + \frac{1}{11} a^{3} - \frac{3}{77} a^{2} - \frac{25}{77} a$, $\frac{1}{77} a^{18} - \frac{30}{77} a^{14} + \frac{1}{11} a^{13} - \frac{34}{77} a^{12} + \frac{2}{11} a^{11} - \frac{24}{77} a^{10} - \frac{2}{77} a^{9} - \frac{23}{77} a^{8} - \frac{15}{77} a^{7} + \frac{31}{77} a^{6} + \frac{37}{77} a^{5} - \frac{36}{77} a^{4} - \frac{32}{77} a^{3} + \frac{16}{77} a^{2} + \frac{17}{77} a + \frac{10}{77}$, $\frac{1}{567292001086300997463473493681859} a^{19} + \frac{2027093471364585316075315230115}{567292001086300997463473493681859} a^{18} - \frac{15095323259247222459187438674}{4688363645341330557549367716379} a^{17} + \frac{504093558017101982269834745485}{81041714440900142494781927668837} a^{16} + \frac{1510547125773940356903872232818}{567292001086300997463473493681859} a^{15} + \frac{32034055538256387111285121800722}{567292001086300997463473493681859} a^{14} + \frac{156637141815954399705466819093597}{567292001086300997463473493681859} a^{13} + \frac{167791839599964873200849660220547}{567292001086300997463473493681859} a^{12} - \frac{138279788235395063199604915656498}{567292001086300997463473493681859} a^{11} + \frac{188254253859752195068105969986605}{567292001086300997463473493681859} a^{10} + \frac{7637577537832622648609984750750}{51572000098754636133043044880169} a^{9} - \frac{176924947841836967912218725533189}{567292001086300997463473493681859} a^{8} - \frac{123956422146438180222192739195721}{567292001086300997463473493681859} a^{7} - \frac{131605658738429211674761836830292}{567292001086300997463473493681859} a^{6} + \frac{224954781373930979462441578927812}{567292001086300997463473493681859} a^{5} - \frac{254064546582644470882992301273638}{567292001086300997463473493681859} a^{4} + \frac{277696948867771449123159680680004}{567292001086300997463473493681859} a^{3} - \frac{86987176694497510311061447935635}{567292001086300997463473493681859} a^{2} - \frac{51185904546370686574469331736121}{567292001086300997463473493681859} a + \frac{59554088532131800799443395032684}{567292001086300997463473493681859}$

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:  \( 36744656987.7 \) (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

\(\Q(\zeta_{11})^+\), 10.10.2670699013250048.2, 10.10.5048580365312.1, 10.10.116117348402176.1

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 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$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
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$$[\ ]^{2}$
23.2.1.1$x^{2} - 23$$2$$1$$1$$C_2$$[\ ]_{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}$