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Properties

Label 20.20.2525688645...8544.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{70}\cdot 3^{16}\cdot 89^{6}$
Root discriminant $104.74$
Ramified primes $2, 3, 89$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T201

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![806404, 0, -4643456, 0, 9347172, 0, -9388512, 0, 5304924, 0, -1756320, 0, 342348, 0, -38928, 0, 2481, 0, -80, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 80*x^18 + 2481*x^16 - 38928*x^14 + 342348*x^12 - 1756320*x^10 + 5304924*x^8 - 9388512*x^6 + 9347172*x^4 - 4643456*x^2 + 806404)
 
gp: K = bnfinit(x^20 - 80*x^18 + 2481*x^16 - 38928*x^14 + 342348*x^12 - 1756320*x^10 + 5304924*x^8 - 9388512*x^6 + 9347172*x^4 - 4643456*x^2 + 806404, 1)
 

Normalized defining polynomial

\( x^{20} - 80 x^{18} + 2481 x^{16} - 38928 x^{14} + 342348 x^{12} - 1756320 x^{10} + 5304924 x^{8} - 9388512 x^{6} + 9347172 x^{4} - 4643456 x^{2} + 806404 \)

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:  \(25256886457092151447506023978226149228544=2^{70}\cdot 3^{16}\cdot 89^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.74$
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:  $2, 3, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{96} a^{12} - \frac{1}{6} a^{11} - \frac{1}{12} a^{10} + \frac{1}{6} a^{9} + \frac{41}{96} a^{8} + \frac{1}{3} a^{7} + \frac{1}{4} a^{6} - \frac{1}{6} a^{5} - \frac{13}{48} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{17}{48}$, $\frac{1}{96} a^{13} - \frac{1}{12} a^{11} - \frac{1}{6} a^{10} + \frac{41}{96} a^{9} + \frac{1}{6} a^{8} + \frac{1}{4} a^{7} - \frac{1}{6} a^{6} - \frac{13}{48} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{17}{48} a + \frac{1}{3}$, $\frac{1}{672} a^{14} - \frac{1}{336} a^{12} - \frac{1}{6} a^{11} - \frac{13}{224} a^{10} + \frac{1}{6} a^{9} - \frac{89}{336} a^{8} - \frac{1}{6} a^{7} - \frac{5}{336} a^{6} - \frac{1}{6} a^{5} - \frac{5}{56} a^{4} - \frac{1}{3} a^{3} - \frac{97}{336} a^{2} + \frac{1}{3} a - \frac{83}{168}$, $\frac{1}{672} a^{15} - \frac{1}{336} a^{13} - \frac{13}{224} a^{11} - \frac{1}{6} a^{10} - \frac{89}{336} a^{9} - \frac{1}{3} a^{8} - \frac{5}{336} a^{7} - \frac{1}{6} a^{6} - \frac{5}{56} a^{5} + \frac{1}{3} a^{4} - \frac{97}{336} a^{3} - \frac{1}{3} a^{2} - \frac{83}{168} a + \frac{1}{3}$, $\frac{1}{4704} a^{16} - \frac{1}{4704} a^{12} - \frac{1}{6} a^{11} - \frac{37}{294} a^{10} - \frac{1}{3} a^{9} - \frac{111}{392} a^{8} - \frac{1}{6} a^{7} + \frac{50}{147} a^{6} + \frac{1}{3} a^{5} + \frac{977}{2352} a^{4} - \frac{1}{3} a^{3} - \frac{1}{98} a^{2} + \frac{1}{3} a + \frac{149}{1176}$, $\frac{1}{9408} a^{17} - \frac{1}{9408} a^{13} - \frac{37}{588} a^{11} + \frac{281}{784} a^{9} - \frac{97}{294} a^{7} - \frac{1}{2} a^{6} + \frac{977}{4704} a^{5} - \frac{1}{2} a^{4} + \frac{97}{196} a^{3} - \frac{1}{2} a^{2} + \frac{149}{2352} a - \frac{1}{2}$, $\frac{1}{766966549440} a^{18} + \frac{2398899}{63913879120} a^{16} + \frac{71995151}{153393309888} a^{14} + \frac{805639391}{383483274720} a^{12} - \frac{1}{6} a^{11} + \frac{2882186633}{95870818680} a^{10} + \frac{1}{6} a^{9} - \frac{14311047697}{54783324960} a^{8} - \frac{1}{6} a^{7} + \frac{778774863}{1966580896} a^{6} + \frac{1}{3} a^{5} - \frac{42258880343}{191741637360} a^{4} - \frac{1}{3} a^{3} - \frac{2016250507}{63913879120} a^{2} - \frac{1}{6} a - \frac{27820558599}{63913879120}$, $\frac{1}{344367980698560} a^{19} + \frac{2867691569}{172183990349280} a^{17} - \frac{35080638365}{68873596139712} a^{15} + \frac{51802048111}{10761499396830} a^{13} - \frac{2648412107063}{28697331724880} a^{11} - \frac{1}{6} a^{10} - \frac{2300996023399}{8199237635680} a^{9} - \frac{1}{3} a^{8} - \frac{1255116570823}{2648984466912} a^{7} + \frac{1}{3} a^{6} + \frac{3722623038049}{10761499396830} a^{5} + \frac{1}{3} a^{4} + \frac{13435268291269}{86091995174640} a^{3} + \frac{1}{6} a^{2} + \frac{13741338641713}{86091995174640} a + \frac{1}{3}$

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:  \( 609905742996000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T201:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 1440
The 13 conjugacy class representatives for t20n201
Character table for t20n201

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.39731038289582358528.1

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

Sibling fields

Degree 10 sibling: data not computed
Degree 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 24 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: 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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.4.8.2$x^{4} + 6 x^{2} + 1$$4$$1$$8$$C_2^2$$[2, 3]$
2.8.31.175$x^{8} + 8 x^{4} + 46$$8$$1$$31$$D_{8}$$[2, 3, 4, 5]$
2.8.31.175$x^{8} + 8 x^{4} + 46$$8$$1$$31$$D_{8}$$[2, 3, 4, 5]$
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$89$$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{89}$$x + 3$$1$$1$$0$Trivial$[\ ]$
89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
89.3.0.1$x^{3} - x + 7$$1$$3$$0$$C_3$$[\ ]^{3}$
89.6.3.1$x^{6} - 178 x^{4} + 7921 x^{2} - 34543481$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
89.6.3.1$x^{6} - 178 x^{4} + 7921 x^{2} - 34543481$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$